Under siege from fellow Democrats, President Biden’s campaign is quietly testing the strength of Vice President Kamala Harris against former President Donald J. Trump in a head-to-head survey of voters, as Mr. Biden fights for his political future with a high-stakes news conference on Thursday.

The survey, which is being conducted this week and was commissioned by the Biden campaign’s analytics team, is believed to be the first time since the debate that Mr. Biden’s aides have sought to measure how the vice president would fare at the top of the ticket. It was described by three people who are informed about it and insisted on anonymity because of the sensitive nature of the information. They did not specify why the survey was…