The Polish international has been in scintillating form since joining Fenerbahce in the summer, contributing to 26 goals from midfield already this season.
Szymanski moved to the Istanbul giants for just under €10m (£8.5m), but Fenerbahce are now looking to roughly treble their money less than a year on, given the form of the 24-year-old.
With a contract until 2027, the Turkish club are in a strong position in the market, and could try to start a bidding war between the three interested parties from the Premier League.
Fichajes claim that Spurs were first to show interest in Szymanski, but have now been joined by Man United and Arsenal, as well as Serie A side Napoli.
The report states that Napoli may be priced out of a move though, with Fener's valuation of €25m (£21.5m) believed to be out of their range, with the Partenopei only willing to spend €18m (£15.5m).
After making his breakthrough at Legia Warsaw as a 17-year-old, Szymanski soon earned a lucrative move to Dynamo Moscow in Russia ahead of the 2019-20 season.
Continued improvement saw Szymanski shine in the 2021-22 season, contributing to 13 league goals, as Dynamo finished third, but the War in Ukraine which started that same year cast doubt over his future at the club.
With Russian clubs banned from European competition, Dynamo were excluded from the Europa League, and Szymanski would join Feyenoord on loan in the summer of 2022.
In the intense atmosphere of De Kuip, Szymanski thrived, helping the club to the Eredivisie title, and an impressive run in the Europa League, reaching double figures for goals that campaign.
The Dutch champions would miss out on signing him permanently over the summer though, as Szymanski opted to move to Fenerbahce along with the likes of Dusan Tadic, Edin Dzeko and Fred.
Given the perilous position of Russian clubs, Fener were able to get their man for a cut price, and could now make a healthy profit should Szymanski leave in the summer.
The midfielder is set to feature for Poland during the upcoming international break, where they have an opportunity to reach the European Championships in the summer.
Clinching qualification could provide Szymanski with the opportunity to showcase his ability on the biggest stage, and may raise his profile and price tag in the process.