Mr. Patrick and former President Barack Obama have been political allies for more than a decade and have remained friendly. In recent weeks, Mr. Obama has told people he thinks highly of Mr. Patrick but that his entry into the race was coming “very late,” according to two people who have spoken with the former president. Mr. Obama sees building a strong organization, especially in Iowa, as a kind of compulsory exercise for a serious candidate, these people said.

Discussing the race with Mr. Patrick, the former president covered the same talking points he had in his conversations with other candidates who have sought his counsel, according to a person with knowledge of their interaction: Campaign outside your political base, stay true to your beliefs and keep beating Mr. Trump in front of mind.

“He didn’t ask for anybody’s blessing,” said Valerie Jarrett, the former senior adviser to Mr. Obama, referring to Mr. Patrick. Ms. Jarrett, who is friends with Mr. Patrick and publicly urged him to run last year, praised him as an “outstanding leader” but stopped short of offering an endorsement — taking care to note that he’s entering “an already strong Democratic field.”

Mr. Patrick’s entry — and his targeting of Mr. Biden — will nevertheless surely test allies of Mr. Obama, as both candidates will claim to be an extension of the former president’s legacy. Mr. Obama, a person close to him said, has spent more time offering assistance to Mr. Biden and his staff than other campaigns, but that was less as a sign of a preference for Mr. Biden’s candidacy than his desire to offer personal support for a man he views with affection.