Rumours began circulating earlier this week that Nottingham Forest, Sheffield United, Bolton and Bristol City are all tracking the 25-year-old.
But while Oztumer’s performances for the Saddlers last season have led to a host of admirers,it is understood none of those four sides are preparing an imminent bid.
Sheffield United are believed to have expressed some interest in the creative schemer but have offered a contract to free agent Jay O’Shea who plays in Oztumer’s favoured number 10 role and impressed for the Blades while on loan from Chesterfield last season.
They also have Mark Duffy and Samir Carruthers in that position.
Nottingham Forest are not believed to be looking for a number 10 with Zach Clough and Ben Osborn among their ranks.
Bolton’s interest is also wide of the mark with the Trotters currently under a transfer embargo.
Bristol City meanwhile are not believed to be interested in the former Peterborough man.
But while a move away from the Banks’s Stadium is not imminent, Walsall are braced for more interest in Oztumer this summer.
Boss Jon Whitney revealed last month the diminutive star is yet to sign an improved contract currently on the table.
And that means the Saddlers hierarchy has to decide whether they cash in on their talisman or risk losing him on a free next summer.