Now Alexander has played down talk, insisting he's not been approached and that he's too busy with his current job to think about Town.
"It's laughable how these things come out," Alexander said. "I've not been approached by Grimsby.
"I've got enough on my plate here to worry about what's going on up there. We've got a big month coming up in the league and all my attentions are on that."
Imps chairman Rob Bradely told the BBC that Alexander was on the verge of signing a new contract and the specualtion was just that, speculation.
"People have put speculation together based on Grimsby losing their boss and Keith coming back to work," Bradley said.
"There was a Grimsby connection in the past but there's nothing in it at all.
"Keith has done very well for us in the past, and as a result we're just dotting the I's and crossing the T's on another contract for him."