The 23-year-old Spaniard spent the second half of last season on loan at Sunderland from parent club Fiorentina.
After impressing at the Stadium of Light, Alonso is hot property and is expected to be the subject of a transfer battle between Sunderland and West Ham whilst his Serie A owners may also yet choose to keep him.
Alonso was a consistent performer during his six months at Sunderland and played a key part in the sides great escape from Premier League relegation as well as featuring in the League Cup final defeat to Manchester City at Wembley.
Despite reports claiming that West Ham are close to a deal on the defender, Poyet remains hopeful of capturing Alonso's signature.
Poyet told Spanish publication MARCA;
"I'm there with 'Marquitos' Alonso, I already had him for six months. We are doing everything possible so that Fiorentina give him to us"
Sunderland are in desperate need to bolster their defensive ranks following the release of Phil Bardsley, Andrea Dossena and Carlos Cuellar.
It's unlikely that Alonso would oppose a move to Sunderland on a permanent basis after settling in nicely on Wearside, but the Black Cats may struggle to compete with West Ham should the London club offer the Spaniard a large wage packet.
West Ham have already rivalled Sunderland once this summer offering a reported £50,000 per-week deal to the out-of-contract Jack Colback before the 24-year-old eventually opted for a move across the North East to Newcastle United.
As it stands the immediate future of the 23-year-old is unknown. Alonso is set to join up with Fiorentina for the beginning of their pre-season and the Serie A club may yet decide that they do not want to sell.