The People says Spurs boss Harry Redknapp has already had one bid of £10m for the 25-year-old knocked back.
But he has been encouraged to improve his offer and hopes an extra £4m will get the deal done.
Buenos Aires-born Osvaldo, who holds dual Argentine and Italian citizenship, has a release clause in his Espanyol contract which would be triggered by a £27m offer. However, sources close to the Catalan club insist business can be done for almost half that amount.
Osvaldo’s agent, Dario Dacoud, said: “The boy is happy in Spain but if a proposition appears which is good for everyone then it would be normal to accept.”