The 27-year-old playmaker, a popular figure among supporters despite fluctuating periods of form during his 18 months at Elland Road, is understood to be wanted by the top-flight outfit in their quest to clinch European qualification in La Liga.
It is believed Saiz, a £3m signing from Huesca in 2017, who has two-and-a-half years left on his contract, is unsettled in England and wants to return to Spain with his pregnant partner.
His departure would be a blow for Leeds, who sit five points clear of the chasing pack in the race for automatic promotion.
Saiz started his first game for six weeks in the 2-1 victory over Queens Park Rangers last Saturday. Bielsa admitted in the wake of Saiz’s return to the starting XI that the midfielder was low on confidence and struggling to replicate his early season form.
Whether Leeds are open to Getafe’s interest remains to be seen, but if Saiz is lured away then Bielsa may be forced into the transfer market.
Izzy Brown, yet to make his debut after joining United on loan from Chelsea last summer when recovering from a serious knee injury, is capable of filling the playmaker role.
But the former Huddersfield Town loanee is a very different player to Saiz, who when on form has proved to a handful for any Championship defence.
The plan was for Brown and Saiz to compete against each other for a place in the team.
Just last week Bielsa all but ruled out any outfield additions in January despite the injury crisis that has left his squad down to the bare bones at times this term.
Only a goalkeeper seemed to be on his wish list after Chelsea loanee Jamal Blackman’s season had been cut short by a broken leg.
Brown will not be involved tomorrow at Bolton Wanderers despite impressing for the Under-23s in the midweek 6-3 victory over Burnley that saw Patrick Bamford, another long-term injury absentee, net a hat-trick.
Bielsa has been happy with the progress made by Brown in his attempts to return to full fitness, but believes more work on building his physical strength is required.
“The sudden movements he can make,” explained the Argentinian, “when he sprints, when he stops, the switch of direction, the short sprints, these are the elements that are needed for a player with a technique like Izzy Brown.
“You need to have a certain level of performance to impose your technical skills. He is working on these small details.”
As for Bamford’s prospects of returning against Bolton Wanderers after three-and-a-half months out, Bielsa added: “He scored three goals (for the Under-23s) and for a No 9 that is a very important thing.
“He is well and he is available for the team. Of course, to reach his previous level he needs more activity and more games. But he is available to start or play as a substitute (tomorrow).”
United have made light of the travails that have seen Bielsa’s squad suffer no fewer than 20 injuries since Adam Forshaw broke down in pre-season.
Four wins in a row have opened up a handy gap on third-placed Sheffield United, who are in action tonight at home to West Bromwich Albion.
Just one goal has been conceded during that run, an impressive feat considering only Pontus Jansson and Barry Douglas of the recognised senior defenders at the club are fit right now.
Bielsa is understandably pleased with how Leeds have hit form heading into the festive season. But the Argentinian has also warned of the need to guard against complacency.
“The only thing I can do is to be happy about it,” he said about the victories over Bristol City, Reading, Sheffield United and QPR.
“And if you take the teams and see how you link different results you would see that you have ups and downs for each team.
“When you win the previous game you take the next game in better conditions but you don’t have any guarantee.
“Sometimes when you have positive cycles your self-esteem is stronger and sometimes you become more relaxed and when you are relaxed you are weaker.”
Bolton are on a poor run, Phil Parkinson’s side having won just once in 17 league outings.
But Wanderers did give leaders Norwich City a huge scare last weekend before Daniel Farke’s side netted a stoppage-time winner through Teemu Pukki.
“They (Norwich) won two games in the last minutes,” Bielsa said when asked about the Canaries keeping Leeds off top spot. “You can evaluate this by saying it is a positive thing and also with the help of luck too.
“We could say the same thing about us because Bailey stopped a penalty that gave us two points, and Sheffield United made a mistake. But we deserved to have those points thanks to the way we played.”