The South African striker wants to see out his contract, which expires in the summer, especially with Ajax in the last 16 of the Champions League.
However, the Daily Mirror reports Ajax would be willing to let the 23-year-old go in January so that they will get some sort of fee.
£1m is thought to be enough for Pearce to get his man, and the Mirror even believes he will try to seal the signing of Pienaar on the very first day of the transfer window.
Pearce watched Pienaar during Ajax's CL clash with Arsenal last week, but faces competition from Celtic and Borussia Dortmund.
Another negative is that Pienaar would miss the majority of January for the African Nations Cup.