Apart from the euphoria which victory brought to the Israelis and defeat causing depression, loss of prestige and honour to the Arabs, the War enabled the once separated populations to create some degree of contact, commercially if not politically or socially. Jews from West Jerusalem visited the Western Wall in East Jerusalam, their most holy shrine as well as the restaurants, cafés and markets while Arab taxi drivers plied the roads in West Jerusalem and elsewhere looking for custom. At a local and intimate level, the war brought about direct and continuous interchange between the Jewish population of Israel and the Arab populations in the West Bank, Jerusalem and Gaza. The barbed wire barriers between Israel and the West Bank and the block wall separating East and West Jerusalem disappeared. This gave opportunities for both confrontation and cooperation between the peoples which they had not had since 1949.
a. Israeli Attempts at Peace Making with its International Neighbours and their Rejection.
It has been suggested by General Odd Bull that if Israel had immediately offered to return the territory to those neighbouring states from whom it had been captured, it could have avoided a continuation of the conflict with its neighbours and achieved "real peace" as well as the consequences of its "occupation" of the West Bank and Gaza The shock of such an offer coupled with the shock of their defeat, according to Bull, (Odd Bull, p.125) might have been sufficient to have terminated the "cycle of arms and counter arms."
This conclusion seems most unlikely and is not supported by the facts:
The United States called upon Israel to withdraw from the conquered territories in return for signed peace treaties.
In response, on June 19, 1967, the Israeli government offered:
to Egypt: the return to its sovereignty over the Sinai Peninsula provided it was demilitarized; and
to Syria: an Israeli withdrawal to the 1922 international border with Syria, provided that the Golan Heights were demilitarized and subject to a commitment that the headwaters of the Jordan in Syria would not be diverted; and
to both Egypt and Syria to conduct separate negotiations regarding the future of the Gaza Strip and the West Bank and a solution to the refugee problem.
The offer, transmitted through the United States, was rejected by Egypt and Syria. (see Moshe Gat,
Although the offer did not mention Jordan or the West Bank, the Israeli government also resolved to open negotiations with King Hussein of Jordan regarding the Eastern border which was problematic for Israel.
While the double shock to the Arabs of the defeat coupled with that of the offer of withdrawal might have brought a positive response if presented to a Western opponent, it is questionable whether pride, loss of face and humiliation would have permitted the Arabs to have accepted such an offer- especially from a dhimmi dominated state. (see Y. Harkabi, Arab Attitudes to Israel, Vallentine, Mitchell, London 1972)
A dhimmi is one who lives in a Muslim society without being Muslim (Jews and Christians) having a lower social, political, and economic status than his neighbour by virtue of his not being a Muslim. The dhimmi was and still is viewed as a second class person; by extreme Islamic fundamentalists he is sub-human. He was “protected” from being killed and treated as an infidel provided he paid a special tax and suffered a number of personal and group indignities. Initially viewed by Muslims with disdain, the dhimmi was later treated with contempt and latterly - especially Jews - with hate.     
The return of territory of itself would not have solved the problem of the Palestinian refugees. Neither would it have satisfied the emerging Palestine Liberation Organisation, established in 1964, which had been given a mandate by the Arab states to act on the behalf of the Palestinian refugees.
b. Israeli Responses
On July 4, 1967, Eshkol appointed a committee to establish contacts within the conquered regions, to report and make recommendations. These recommendations were unanimous:
"[Establish and maintain] intensive political activities, to reach a peace agreement with Jordan.
In the absence of an immediate peace agreement with Jordan, Israel will continue to administer the West Bank "as a separate administrative and economic unit", "a civilian regime with emergency powers". IDF will control the new borders while the domestic security will be the responsibility of the Police. "A special minister will rule the West Bank in the form of a Canton. A small Israeli group of personnel will deal with state policy level while local domestic topics (municipal, etc.) will be handled by local Arab officials.
An immediate search for a comprehensive solution of the Refugee Problem (that time there were 23 refugee camps in the Gaza Strip and the West Bank), either in the framework of the Peace Agreement or by Israel, recruiting the help of the International Community.
Two tracks - the Jordanian on one hand and the Israeli-Palestinian - will be taken simultaneously "since they are not necessarily interdependent." 
In Jerusalem especially, the ability for Jews to visit their holy places also gave them the opportunity to visit the Arab markets and restaurants. Fairly soon after the 1967 War, Israel began to see the employment of Arab workers, within Israel, especially in the building trade. The concern of both peoples seemed to be centred on the restructuring of their lives.
Having been denied access by Jordan to its most holy places in Jerusalem and all the Jewish places of worship having been destroyed by Jordan while under its control, Israel was not about to risk the physical surrender of its most precious symbols of its Jewish identity to those who believed that the Jewish state was illegitimate and who were capable later of replacing an ephemeral "peace" to one of physical violence.
There was, therefore, a consensus in Israel's government that East Jerusalem should be annexed.
c. Palestinian Views and Israel’s Reaction
At the political level, selected Israeli military reservists were directed to undertake a fact finding mission - to make contact with local leading Palestinians in order to assess their political opinions.
“A clear picture emerged from their conversations: The Palestinian Arabs-except for a minority with special interests did not want to return to Jordanian rule. They suffered from economic discrimination designed to favour the East Bank Jordanians against the West bank Palestinians. More than anything they wanted to be free to shape their own future. They were therefore prepared to strike a deal with their latest masters…in return form an independent state or entity on the West Bank and in Gaza they were willing to sign a formal peace treaty with Israel and co-exist with us in every way. This was in stark contrast to the conventional position of the Arab states.
Those of us who had been in daily touch with the Palestinian groups, felt that we had before us an opportunity to reach an agreement with representative Palestinians which should not be missed even if it meant giving up some of the land we considered to be ours” (see David Kimche, The Last Option, Weidenfeld & Nicholson, London, 1991, pp 241-248) (gma emphasis)
 Moshe Gat Britain and the Occupied Territories After The 1967 War, 10, MERIA J. Dec.2006
 Moshe Sasson, Levi Eshkol – A Man of Peace