Lita has been courted by both Queens Park Rangers and Swansea City both promoted from the Championship last season in recent months.
But the Middlesbrough striker, who has 12 months remaining on his contract at the Riverside Stadium, is in no rush to leave after enjoying a new lease of life under Tony Mowbray.
While QPR's interest has waned in recent weeks, the Swans remain keen on landing the former Reading man and have been in touch with Boro.
Swansea are aware that Middlesbrough have to sell at least one player if Mowbray wants to start to bring in his own signings, with Frenchman Malaury Martin the only senior arrival so far.
Marvin Emnes remains Swansea's main target from Middlesbrough, but they realise that Mowbray is intent on retaining the services of the Dutchman unless their valuation is met.
That could mean Swansea turn their attentions once more towards his team-mate and Lita would be a lot cheaper than Emnes. However, Lita has hinted to those close to him that he would be prepared to consider a new contract offer to stay.
Lita played more than an hour of last night's goalless draw with Antalyaspor in the Slovenian town of Dravgrad.
It was the last of three friendlies Mowbray lined up for his squad during their eight-day stay at a training camp in Slovenia.
While the tour did not end in a victory, the draw meant that they maintained an unbeaten start to pre-season.
Martin, signed for free after his release from Blackpool, made his first start for the club in the centre of midfield.
Merouane Zemmama, who had not figured in Sunday's win in Croatia because of visa problems, joined him in the team.
Mowbray was also impressed by the performance of young goalkeeper Connor Ripley.
The son of former Middlesbrough winger Stuart, Ripley had to start because of an injury to Danny Coyne, who picked up a dead leg in his last outing.
Middlesbrough: Ripley; McMahon (Hoyte 46), Williams R, McManus (Main 62), Bennett (Reach 62); Robson (Kink 62), Smallwood (Franks 62), Martin (Bates 46), Zemmama (Grounds 46); Lita (Miller 62), McDonald (Williams L 46). Subs (not used): Coyne (gk)