The Daily Mail says the midfielder has been frozen out at German club Stuttgart having played just nine minutes of football all season.
Allardyce is desperate to bolster his squad ahead of their fight to stay in the Premier League. And the Rovers boss has earmarked the 31-year-old attacking midfielder as one of the men he wants to bring in.
The ex-Bolton boss wants to take Basturk on loan with a view to a permanent deal in the summer.
Basturk has been told by Stuttgart manager Christian Gross that he does not fit into his playing system.
A Stuttgart source said: "There's a good chance Basturk could be heading to the UK in this transfer market."