Speculation linking Costa with a move to China seemed to result in a loss of form in January and fresh links could also be behind his recent slump as he has gone five Premier League games without scoring.
Tianjin are reportedly ready to make Costa the best-paid player in the world, with a contract worth £650,000 a week after tax.
Talks between Costa and his advisers ensured he remained in place at Chelsea until the end of the season, but it appears that the 28-year-old could now be close to agreeing a move to China's Super League.
Diego Costa remains a £620,000-per-week NET target for the Chinese side, who are managed by former World Cup victor Fabio Cannavaro.
Nevertheless, according to The Daily Mail, Tianjin are still working on a £620,000-a-week deal to make their biggest signing yet, with owner Shu Yuhui not shy in discussing the ambitions of the club and how they want to see themselves as genuine competition for the top European outfits in the transfer market. "And we have already made contacts with several players".More news: Westbrook vs Harden the highlight when NBA playoffs open
And Yuhui suggested that his club will continue to try and tempt top players to join his club, saying: "We have high standards [in buying foreign players] and those players prefer top clubs such as Bayern Munich and Real Madrid".
Costa is contracted to Chelsea until the summer of 2019, with the striker earning around £150,000 (HK$1.5 million)-a-week with the London club.
"We are literally grabbing players with Real Madrid and Bayern".
The Chinese Super League season began in March and ends in November, with a mid-season transfer window open from June 19 to July 14, during which time a massive offer is expected to come in for Costa.