“How well did the president respond to a new and totally surprising set of circumstances and did he keep Americans safe? That’s the question,” said Senator Patrick J. Toomey, Republican of Pennsylvania, who reluctantly supported Mr. Trump in 2016.

Mr. Biden and Mr. Trump are facing a combination of conventional and wildly atypical challenges as they attempt to reposition themselves for the general election. For Mr. Biden, the traditional tasks of healing primary-season wounds and building up stronger campaign machinery have been greatly complicated by the context of the virus. He installed a new campaign manager, Jennifer O’Malley Dillon, and his aides have been recruiting a more robust staff, but many of the usual functions of a campaign — including voter outreach and fund-raising — have been slowed or halted by the quasi-national lockdown.

His efforts to smooth divisions within the Democratic Party have taken place over the phone or in public forums like television interviews. He has spoken by phone to a number of his former primary rivals, including Mr. Sanders and Senator Elizabeth Warren, endorsing several key liberal priorities aimed at securing their energetic support. On NBC’s “Today” show on Tuesday, Mr. Biden said he wanted Mr. Sanders to be “part of the journey” for his campaign and eventual administration.

Even the usual process of auditioning vice-presidential candidates has been upended by the virus, since it is not possible under the circumstances for Mr. Biden to hold in-person meetings with a range of options and go out for a day or two with each of them on the campaign trail. Still, Mr. Biden has said that he would begin the process of seriously vetting potential running mates within weeks.

Democrats are fretting, as they often do, that Mr. Biden has been diminished or at least obscured by this crisis. But his performance, at least until the virus subsides, is far less important to the outcome of the election than whether Mr. Trump is seen as handling this crisis well in the crucial weeks ahead.

Mr. Trump, for his part, has been experimenting with a number of different campaign messages to replace the one he had counted on using: one that combined a promise that re-electing him would bring four more years of economic prosperity, with a warning that a far-left Democratic nominee would imperil that trajectory.

With the economic argument off limits, at least for now, and the Democratic Party’s left wing humbled in the primaries, Mr. Trump has been floating a number of alternative arguments for his candidacy in the White House briefing room, to uncertain effect.