Donald J. Trump is publicly making the case that Roy Moore, the controversial former Alabama judge, could cost Republicans a Senate seat in an overwhelmingly conservative state if he chooses to run again.

One person he hasn’t convinced is Mr. Moore himself.

“There’s a lot I have to offer,” Mr. Moore said in an interview Wednesday, adding that he would make up his mind by the end of June whether to challenge Doug Jones, the Democrat who narrowly beat him in 2017. He added, “Everything seems to be very favorable.”

President Trump took to Twitter early Wednesday to warn that if Mr. Moore was the Republican nominee in 2020, he would most likely lose again to Mr. Jones, considered the Democratic Party’s most vulnerable incumbent up for Senate re-election next year.