Time is short for Massachusetts Republicans, and the list of potential GOP candidates to run in the upcoming Senate special election is increasingly small.

On Monday morning, former Gov. William F. Weld, a Republican, said he would forgo a bid to fill the seat formerly held by Secretary of State John Kerry.

“While I am grateful for the kind expressions of support and encouragement which I have received, I will not be a candidate for United States Senator from Massachusetts in the special election this year,” Weld said in a statement.

Weld joins a growing list of Bay State Republicans who have rejected a Senate campaign. To make matters more daunting for the local GOP, any potential Republican candidate must collect 10,000 voter signatures by the end of the month.

Tagg Romney, the son of 2012 GOP presidential nominee Mitt Romney, won’t run, ABC News reported. Over the weekend, former state Sen. Richard Tisei said he wouldn’t mount a bid either.

Bay State Republicans don’t expect former Lt. Gov. Kerry Healey to run, leaving a slim cast of GOP names pondering a Senate effort. They include state Rep. Dan Winslow, who is expected to make an announcement Tuesday, and former Navy SEAL Gabriel Gomez.

Reps. Stephen F. Lynch and Edward J. Markey are the only declared Democratic candidates. The Boston Globe reported over the weekend that Middlesex County District Attorney Gerard T. Leone might also jump into the Democratic primary.

The primary election will be held April 30, and the general will be held June 25.

Roll Call rates the race as Likely Democratic.