The 23-year-old was drafted to his current team at the start of 2022, and following a season in the MLS Next Pro, he played a significant role in Columbus' MLS Cup success in 2023.
Schulte kept nine clean sheets in 37 regular and post-season matches as the Crew defeated Los Angeles FC 2-1 in the MLS Cup final, and he has begun the 2024 campaign with two clean sheets from five top-flight and CONCACAF Champions Cup matches.
A one-cap USA international, the 6ft 4in stopper made his Stars and Stripes debut in a 1-0 friendly loss to Slovenia back in January, and his contract with the Crew only runs until December, albeit with the option of an extra year.
According to the Daily Mail, Schulte's displays have attracted the attention of Arsenal and Man United, both of whom could be in the market for new backup goalkeepers in the summer window.
The Gunners are likely to lose current number two Aaron Ramsdale, who spent two seasons as Mikel Arteta's undisputed number one, demoting Bernd Leno to the bench and also restricting the minutes of Schulte's compatriot Matt Turner.
However, Brentford loanee David Raya has now become Arsenal's first choice, and the Spaniard - who saved two penalties in Tuesday's Champions League last-16 success over Porto - will officially join permanently this summer.
Ramsdale's only Premier League starts since the September international break have come against Brentford, whom Raya has been ineligible to face, and he was guilty of a horrific blunder for a Yoane Wissa goal last weekend.
The Englishman redeemed himself with two crucial second-half saves in a 2-1 win, but his otherwise limited game time will harm his chances of a Euro 2024 call-up, and he is expected to seek an exit this summer, despite only extending his deal until 2026 last year.
Meanwhile, Man United added both Andre Onana and Altay Bayindir to their ranks last summer, and despite scathing criticism at the start of the season, the former has been Erik ten Hag's first choice in all competitions.
Bayindir's only appearance for the Red Devils so far came in a 4-2 FA Cup win over Newport County in January, when Onana was away at the Africa Cup of Nations, and he is already being lined up for a move back to Turkey.
Furthermore, 37-year-old Tom Heaton - Ten Hag's third-choice stopper - is out of contract this summer, and the Red Devls sold Matej Kovar to Bundesliga leaders Bayer Leverkusen last summer.
The report adds that Premier League teams have been "extensively" scouting Schulte over the past year, but Arsenal and Man United are not the only top-flight sides who could be in the market for new number twos.
Liverpool's Caoimhin Kelleher - currently shining in the absence of Alisson Becker - will be the subject of summer bids, while Chelsea duo Robert Sanchez and Kepa Arrizabalaga are both facing uncertain futures too.
Manchester City are discussing a new deal with Ederson's deputy Stefan Ortega, but 38-year-old Scott Carson is also due to become a free agent at the end of the campaign.