“We think Congressman Kennedy should run for the United States Senate not simply to oppose any person or because his last name is Kennedy,” the group states on its website. “The congressman should run because our country is vulnerable, and he has demonstrated that he has the energy, courage, and progressive ideas to fight for the Commonwealth and put our nation on a more just course.”

Mr. Kennedy delivered the Democratic response to President Trump’s State of the Union address last year. But he has otherwise not cut a high profile in Washington since being elected in 2012, despite his political pedigree. But in recent days, he has started to speak out more, writing an opinion article this week about his grandfather for The Washington Post and sitting for an interview on the podcast of David Axelrod, the former adviser to President Barack Obama.

Mr. Kennedy’s moves have not gone unnoticed by Mr. Markey, who won a special election in 2013 to fill the seat vacated by John F. Kerry when he was appointed secretary of state. Mr. Markey has $4 million on hand, has hired a pollster and, moving to amplify his liberal credentials, has worked with Representative Alexandria Ocasio-Cortez of New York on the Green New Deal.

He has also secured the support of Senator Elizabeth Warren of Massachusetts, who has taped a video in support of his re-election, according to two Democrats. (Mr. Markey, who was already facing two lesser-known primary race challengers, endorsed Ms. Warren’s presidential bid when she entered the race this year.)

Mr. Markey has sought to flush out his would-be rival, releasing a list of endorsements from the state’s congressional delegation this week to The Boston Globe — a roster that notably did not include Mr. Kennedy. Also absent from the list was Representative Ayanna S. Pressley, the Boston Democrat who won her own primary race last year against a longtime incumbent and is widely thought to also be eying a Senate bid, whether if against Mr. Markey or if Ms. Warren were to be elected president.

With a host of other Massachusetts Democrats, including the attorney general, Maura Healey, also weighing future bids for the Senate and governorship, Mr. Kennedy’s bid could change some of those calculations.

Mr. Markey and Mr. Kennedy have not seen each other of late, but they did talk last week on the phone, according to a Democrat familiar with the conversation. Mr. Markey called to offer his condolences to Mr. Kennedy in the days after Saoirse Kennedy Hill, Mr. Kennedy’s cousin, died of a drug overdose in Hyannis Port, Mass.

The Senate race did not come up, but the two Democrats concluded their conversation by agreeing that they should meet soon.