He is out of contract at Bayern Munich at the end of next month and has accepted he will not be staying at the Allianz Arena.
The 32-year-old has options to stay in the Bundesliga, with Hamburg and Werder Bremen keen, while there is also significant interest in Spain, where Valencia and Sevilla are prepared to make an offer.
But Klose is tempted to play in the Premier League and Sunderland are hoping to fend off increasing competition to land his goalscoring quality.
The Germany international is believed to have suggested to those close to him that a London club would be his preferred destination.
But with a lack of takers, despite the likes of Arsenal and Tottenham expressing tentative interest, Sunderland manager Bruce is keen to do business.
One stumbling block would be his reputed £100,000-plus weekly wage, but with no transfer fee Sunderland are keen to at least have the opportunity to tempt him with a switch to Wearside.
Klose also linked with Everton, QPR and Norwich has had limited playing time with Bayern this season, but his goalscoring record has been exceptional particularly for his country.
He has scored 61 times in 108 appearances for Germany and scored 14 World Cup finals goals, a record only bettered by Brazilian Ronaldo.
Bruce is having to put plans in place now in an attempt to ensure he does not encounter the same problems as the last five months.
At the beginning of last season, he had arguably one of the best strike-forces in the Premier League. Twelve months on, he is preparing to start from scratch.
The boss had Darren Bent, Danny Welbeck, Fraizer Campbell and Asamoah Gyan at his disposal. But he heads in to tomorrow's visit of Wolves without a single senior striker, forcing him to play the likes of Stephane Sessegnon and Ahmed Elmohamady up front.
He said: You don't have to be a rocket scientist to realise where we are drastically short. That has got to be addressed.
No disrespect to the likes of Daryl Murphy, Anthony Stokes, Dwight Yorke and Kenwyne Jones.
I look at that and then the transformation to Bent, Welbeck, Gyan and Fraizer, it's light-years away from them.
Unfortunately, Danny's done that well he's going back to Man United and the loss of Darren and Fraizer to injury, has left us with Gyan.
That is the biggest frustration for any manager, because the top end of the pitch is more important and we thought we had that boxed off for years but now we have to start again.
Bruce had to wait until the last day of the transfer window to complete the signing of Gyan last summer, and admits he may have to be patient in his quest for signings once more.
In my experience, there's nothing that usually goes on in June, Bruce said.
We got Bent late on as well so I wouldn't be surprised if it takes until the season's about to start before we act.
We have irons in the fire.