President Reagan welcomed Australian Prime Minister Robert Hawke to the White House yesterday, and afterward the two men said they had established a personal relationship that should maintain the traditionally close ties between the two countries.

Hawke, making his first visit here since becoming prime minister in March, said as he left the luncheon meeting that there is no country the United States "will be able to rely on more than Australia."

Reagan called Australia "a key ally on whom we can count."

The warm exchange made clear that despite the gulf between the political philosophies of Hawke, who leads the left-of-center Labor Party, and the conservative Reagan, there are no problems likely to cause serious tensions in U.S.-Australian relations.

The relationship is not without its occasional bumps. The Australians think that some U.S. trade and agricultural policies hurt their exports; and the Labor Party has made noise, potentially disturbing to Washington, about joint intelligence facilities in Australia and about resuming aid to Vietnam.

However, a senior U.S. official who briefed reporters after the meeting said that these subjects were discussed only in the most general way. The official, who spoke on condition he not be identified, said Reagan and Hawke "really hit it off," and he stressed that "even where there are differences, there is the strongest possible basis for working them out."

On the Vietnam issue, the official reiterated that the United States opposes aid to Hanoi as long as Vietnam occupies Cambodia. The official described the U.S. position as being in line with that of the Association of Southeast Asian Nations (ASEAN) and said the Australians were told that "we consider support for the ASEAN countries the key to finding a solution" to the Cambodia question.

Asked about U.S. intelligence-monitoring and communications activities in Australia, the official noted that the United States does not have bases there but "joint facilities that serve both our interests."

He said there had been no discussion of a proposal to station an Australian liaison officer here to help monitor the intelligence collected by the United States in Australia. However, the official added, that subject might come up in talks between Hawke and Defense Secretary Caspar W. Weinberger.

The State Department announced yesterday that Secretary of State George P. Shultz will leave June 23 for visits to Manila, Bangkok, New Delhi and Pakistan, returning to Washington July 5.

In Bangkok, Shultz is to attend a meeting of the Association of Southeast Asian Nations; and although there has been persistent speculation that he will revisit the Middle East while en route home, a department spokesman reiterated that there are no such plans.