Spanish side Espanyol had been leading the chase for Barrera, who impressed at this summer's World Cup finals, and opened talks with his club UNAM Pumas last month.
But there is now strong interest emanating from Britain with West Ham, Everton and Celtic all reported to be keen, along with Italian side Lazio.
Pumas have now confirmed that they are considering three offers - from West Ham, Everton and Espanyol - with one bid standing out.
Reports in Mexico claim that the best offer is from Upton Park and they are now in pole position to claim his signature.
"We don't want to be making announcements when it's not the correct moment. When everything is OK we will make an announcement," said Pumas director of sport Mario Trejo.
"It's very possible that between Thursday and Friday we will have something more serious to say.
"If there is an important announcement to be made, the president will be in charge of letting everyone know.
"I must admit that after analysing the offers of Espanyol, West Ham and Everton, there is one that has convinced us more than the rest.
"We can't talk too much about it because then everything gets complicated and as we are very near to reaching an agreement we don't want to spoil the negotiations."
West Ham have also been linked with Barrera's team-mate Efrain Juarez, but Trejo says they are yet to receive a bid for him.
"Juarez? Nothing, nothing official or written, not yet," he added.