The Premiership’s top scorer has agreed a pre-contract with Dundee United.
Watt was dropped for the final league game before the winter break and Alexander revealed “the timeline of when he leaves is not 100 per cent”.
He told Press Box Podcast: “It's not even been talked about, to be honest.
"We've been told that he's signed a pre-contract at Dundee United and that's as far as it goes.
"It's quite an unusual situation for me, to be honest, because I've never really experienced this before because obviously it's my first stint managing in Scotland.
"The pre-contract, signing for another club in your own league and still playing for the other club, is quite an unusual and unique set of circumstances, so I'm still getting my head around that scenario.
"He will leave but the actual timeline for when he leaves hasn't been decided because we haven't heard from Dundee United in terms of a bid or anything.
"If he's here until the end of the season, we will enjoy having him with us from now until then."
Dundee United boss Tam Courts is keen to add the striker to his squad in January.
The Tangerines are one of the lowest scorers in the league with St Johnstone the only team to have scored fewer.
According to the Daily Record, the Tannadice side are willing to offer players and cash to push the deal through this month.
Alexander compared the situation to Declan Gallagher last season when the centre-back signed a pre-contract agreement with Aberdeen.
Once he remained after the window the team “cracked on”.