The Colombia international has impressed at the Allianz Arena since joining on a two-year loan deal last July.
Bayern chairman Karl-Heinz Rummenigge confirmed last month that they have the option to sign James on a permanent deal for €42 million, although he stressed there was no rush over triggering the clause.
James, who scored his seventh goal and got his 11th assist in the Bundesliga in Saturday's 3-1 win over Cologne, admits he would be delighted to stay in Bavaria for the long term: "I'm doing very well.
"It's a good league here in Germany, the stadiums are full. I came here with a different perception.
"I'm fine, I hope I stay here. I feel very good."