Sky Bet suspend betting on Radamel Falcao's next club after seeing him backed at as short as 1/3 to join Real Madrid.
Morgan Schneiderlin has appealed to Southampton fans after a thwarted attempt to leave St Mary's.
As things stand, the Germany international is due to become a free agent at the end of the season.
It has been suggested that, as he will turn 36 over the summer, Klose will look to take on a new challenge when his deal in Italy is up.
A switch to the American MLS has been mooted, as has a spell in the Premier League.
Lazio, though, are refusing to give up on their efforts to tie Klose to fresh terms.
President Claudio Lotito told Tuttomercatoweb: "There is no problem on our part.
"I think Klose has been happy in Rome. I think we can soon find a solution in keeping him."