The Argentine is subject to a lot of interest around Europe, particularly in the Premier League as Manchester United, Liverpool and Tottenham have all been linked with the 25 year-old.
However, it is understood that Fulham are the latest club to express an interest in the former Fiorentina man after they apparently had a €12 million [£10.5m] bid rejected, with the club labelling the offer as "unsatisfactory".
Espanyol, like many Spanish clubs, are believed to be in a less than enviable position financially, however it is understood that they are nevertheless holding out for a bid closer to €20m [£17.5m], the sort of money the Cottagers are unlikely to be able to stump up.
Another potential stumbling block is the claim of Osvaldo's agent Dario Decoud that the former Italy Under-21 international is happy in Catalunia, suggesting he will only move for a considerable offer.
Decoud said: "Osvaldo is happy with Espanyol. Only a big offer could change his mind.”
This season, Osvaldo enjoyed his most fruitful season in front of goal, scoring 13 times in 24 league appearances.