Spurs boss Andre Villas-Boas has been keeping tabs on the club's Italian international striker as he looks to add another top centre forward to his front line, having recently signed Roberto Soldado from Valencia.
But Premier League rivals Southampton are odds on favourites to land the 27-year-old striker having had an offer in the region of £15m accepted by Roma.
However, it's not over for Spurs and the White Hart Lane outfit are monitoring the situation closely.
Osvaldo is reportedly unsure on whether a switch to the south coast club is in his best interests, and is stalling on Saints' proposal in the hope that a better offer is forthcoming.
The controversial frontman has been linked with the likes of Fulham and Benfica, and Osvaldo would prefer a move to London.
Fresh speculation suggests Southampton's executive chairman Nicola Cortese has set Osvaldo a deadline to make a decision on his future - reported to be Friday (today).
The Saints have offered the ex-Espanyol striker - who played under Mauricio Pochettino during their spell together at the Spanish club - a four-year deal with a salary of £2.1m a season in a bid to entice him to the Premier League.
Whilst Tottenham look like they are going to miss out on Osvaldo, chairman Daniel Levy is still in talks with Roma regarding the potential sale of Erik Lamela.