If he is to compete in New Hampshire, Mr. Patrick would have to file there this week: The deadline to appear on the state’s ballot is Friday.

A Quinnipiac poll released on Monday showed four candidates bunched together at the top in New Hampshire, with Mr. Biden at 20 percent among likely primary voters, Ms. Warren at 16 percent, Mr. Buttigieg at 15 percent and Mr. Sanders at 14 percent. (The margin of error was plus or minus 3.8 percentage points.)

A possible Patrick candidacy could complicate the strategic assumptions for a number of candidates, including the two leaders in most polls, Mr. Biden and Ms. Warren. Mr. Patrick could threaten the former vice president’s support from black voters and also make inroads in New Hampshire, where Ms. Warren is counting on a strong performance in the southern part of the state that borders Massachusetts.

He and Ms. Warren have had an amicable, if not personally close relationship; when she was asked at an event last week to name African-Americans she’d want in her cabinet, she included Mr. Patrick.

He is close to former President Barack Obama, as well as some of his top advisers, and has previously discussed a potential White House bid with the president. Late last month, Mr. Patrick was with Mr. Obama for a meeting of the Obama Foundation in Chicago.

Were he to run, the former governor may find it difficult to create a full-fledged campaign organization so late in the process. Two of his longtime Massachusetts aides are already committed in 2020: Doug Rubin is working for businessman Tom Steyer’s presidential campaign and John Walsh is overseeing the re-election of Senator Ed Markey, who is being challenged by Representative Joseph P. Kennedy III.

According to Massachusetts Democrats, though, Mr. Patrick has taken steps toward building a staff.

He asked one former adviser, Jennifer Liu, on Monday if she was willing to oversee his fund-raising. Ms. Liu worked for his political arm when he was governor and was recently laid off from Senator Kamala Harris’ campaign, where she had been a top finance aide.