The Spaniard, who has previous Premier League experience with Bolton, left the Trotters in the summer but has found first-team opportunities limited in Serie A.
"I can confirm Marcos will be with us for the rest of the season on loan, which is something very important for us," Poyet told the media on Monday.
"I was always looking for a player at left-back or on the left side so I'm delighted he's here.
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Alonso is not intended to be the Black Cats' only January acquisition, says the Uruguayan, though he is not sure of whether or not he will be able to sign the "three or four" players that he wants.
"The problem with putting a number on players is that I would like maybe three or four but it's always so unpredictable," Poyet remarked. "The reality is [that] it's almost pointless saying so at the outset because things can change so much.
"Initially, though, three or four is the idea."
Meanwhile, the Sunderland boss says that there is no sign of any players leaving the Stadium of Light mid-season.
"We haven't had any offers for our players, as things stand, and any clubs who are interested will need to make an offer because we are not going to advertise them," he continued. "Every single player here is under contract as things stand and you need to respect that.
"We won't be throwing names out publicly [but] we will consider what interest we get."