Hagi, 24, has been on the books at the Light Blues since 2020 and the Romanian played a key role in Rangers winning the Scottish Premiership title under Steven Gerrard.
He has struggled with injury of late however and is now down the pecking order under Gers boss Michael Beale, with an exit expected.
The Romanian attacker is frustrated at the lack of playing time he is getting at the Glasgow giants and is keen on leaping overboard this summer.
Beale expects Hagi to move on and now the midfielder has an option in La Liga, according to Spanish outlet Relevo.
Spanish side Alaves have approached Rangers to take Hagi on a loan deal.
The loan deal would not contain any option for Alaves to sign Hagi on a permanent basis and it would be a simple loan.
Moving to La Liga is likely to be an attractive prospect for Hagi, who could be facing the likes of Barcelona and Real Madrid.
