Poyet revealed on Monday that the former Bolton Wanderers left-back will return to the Premier League on a six-month deal when the transfer window opens on Wednesday after failing to establish himself in the Fiorentina starting line-up.
"I can confirm Marcos is going to be with us for the rest of the season," said the Uruguayan coach.
"I was looking for a player in that position on the left side and I hope he will help us and everybody will like him.
"I am delighted that everything is agreed."
Alonso joined the Florence club from Bolton in May, but has made only three Serie A appearances for Vincenzo Montella's side.
The Spaniard started his career at Real Madrid before moving to Bolton in July 2010 and made nine Premier League appearances for the Lancashire club before they were relegated to the Championship.
Poyet also confirmed that his first January recruit will be available for the FA Cup third-round clash with Carlisle United on Sunday.
Sunderland are bottom of the Premier League, but the Wearside outfit are unbeaten in their last five games in all competitions ahead of their crunch clash with fellow strugglers Aston Villa on New Year's Day.
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