Watt had already agreed a pre-contract with United and will now move to Tannadice this week.
The 28-year-old has netted 10 times this season and is the cinch Premiership's top goalscorer.
"We had a conversation on Saturday after we trained, me and Tony, and over the weekend the club have agreed a fee which we find acceptable. That is getting officially done as we speak.
"Everything we do is for the best of the club and the team. We knew of the potential situation in the summer, the way it was going, and we decided on a course of action which we would be content with."
Alexander is not set to spring into action to replace Watt having already signed attacking player Ross Tierney and tied up a permanent deal for Jordan Roberts this month.
Motherwell have Kevin Van Veen and Connor Shields as centre-forward options while Kaiyne Woolery was on form out wide before the winter break, with Justin Amaluzor also competing for a place.
"We are not looking for a direct replacement for Tony," said Alexander ahead of Tuesday's game against Ross County in Dingwall.
"We feel we have the players already here at the club to keep the results going.
"We feel we have some real good options in those forward areas and we feel how we play we will continue to score goals and win games, and each individual adds into that."
Alexander has also brought in Celtic midfielder Liam Shaw on loan this month with Robbie Crawford moving to Partick Thistle.
"I'm quite happy with what we have got," he said. "We have competition in all positions.
"We haven't stopped looking, we are always looking. I know the window goes from there to there but in between there is a lot of work, a lot of analysis on players.
"We understand we can't be over-reliant on any individual so we have always got counter-targets for any issues or players leaving.
"We feel we constantly have to try to improve. Even though we are not looking for a position, if someone comes along who we feel can improve us, we look to do something.
"That will continue to be the case. We always have players identified that we can hopefully bring in when the time is right."