The 18-year-old has emerged as one of the standout prospects in the MLS this season, the teenager already holding down a regular spot between the sticks for his current club.
Although Chicago Fire sit at the bottom of the Eastern Conference, Slonina has conceded just 20 goals during his 14 starts during the 2022 campaign.
Having only extended his contract until the end of 2023 in April, neither the player nor Chicago Fire are seemingly in any rush to part ways at this stage of his development.
However, his representative Jaime Garcia has told MLSsoccer.com that the youngster intends to sign for Real Madrid should a deal be reached with the recently-crowned Champions League winners.
Garcia said: "It should be an honor for Chicago, and MLS, to have Real Madrid take notice in one of their players. We are hoping Chicago can find an agreement with Real Madrid to realize Gabriel's dream of playing for this club.
"With respect to other clubs and other offers - we've received many and we are gracious to those clubs - but you cannot say no to Real Madrid."
Chelsea and Wolverhampton Wanderers have both been heavily linked with a move for Slonina in recent months, as have Bundesliga champions Bayern Munich.
In the case of Chelsea, it has been claimed that the Blues were contemplating a formal approach for Slonina before the government imposed sanctions on the club due to Roman Abramovich's links with Vladimir Putin.
Although the Premier League giants can now function normally under a consortium led by American Todd Boehly, it appears that they have missed out to Real.
Meanwhile, Wolves also eyed Slonina as one for the future, their current first-team number one role being locked in with Jose Sa between the sticks.
However, there is a growing feeling at Molineux that more younger players will be signed in the future, Wolves seemingly acting as a stepping stone for prospects who may be sold for a profit in the years that follow.
Slonina would have fitted into that bracket, albeit as a number two to Sa, but it appears that a switch to the Bernabeu is his only intention when departing Soldier Field.