The discussion with Mr. Morrison shows the extent to which Mr. Trump views the attorney general as a crucial partner: The president is using federal law enforcement powers to aid his political prospects, settle scores with his perceived “deep state” enemies and show that the Mueller investigation had corrupt, partisan origins.

The Justice Department inquiry and a parallel but unconnected effort by the president’s personal lawyer Rudolph W. Giuliani represent a kind of two-front war. Mr. Trump has said that Mr. Barr can help him validate his 2016 electoral victory, and Mr. Giuliani has been trying to unearth damaging information about former Vice President Joseph R. Biden Jr. in anticipation of the 2020 race.

As part of his efforts, Mr. Barr in recent months has asked the president to facilitate communications with foreign officials and has made at least one trip to Italy to secure cooperation, according to a department official. The inquiry is examining American intelligence and law enforcement activity around the Trump campaign and whether it was legal.

Mr. Trump initiated the discussion with Mr. Morrison in recent weeks explicitly for the purpose of requesting Australia’s help in the review, according to the two people with knowledge of the discussion.

In making the request — one of many at Mr. Barr’s behest — Mr. Trump was in effect asking the Australian government to investigate itself. F.B.I. investigators began examining Trump ties to Russia’s 2016 election interference after Australian officials reported that Russian intermediaries had made overtures to Trump advisers about releasing information that would damage Hillary Clinton’s campaign.