The Spanish giants went for the Argentina striker when he left Manchester United for his controversial trip across town to City 17 months ago and they have been alerted by his unhappiness at Eastlands.
Bernabeu boss Mourinho reiterated this weekend that he is short of options up front, with uncertainty around Gonzalo Higuain's injury only adding to the problem.
Danger man: Tevez (right) scores against Chelsea at Eastlands earlier this season
'Signing players is not an issue for me to deal with but I already said at the beginning of the season it would be difficult with only (Karim) Benzema and Higuain,' said Mourinho.
'If I have to go through the rest of the season only with Benzema it will be much more difficult. But this is something you have to ask the director general and he can tell you the club's position.'
Tevez does not naturally fit the kind of profile that Mourinho has been looking at, but he has many admirers among the Bernabeu hierarchy.
The nine-times European champions will first want to know Tevez's intentions before making a move. It is unclear whether the 26-year-old would be keen to move to a Latin country like Spain or if he only has his heart on return to South America.
Worthy stand-in: Benzema scores against Auxerre in the absence of Higuain
The striker is missing his two daughtersin Argentina but recently suggested it would be out of the question to play in his homeland again. While Boca Juniors were his first professional club and first love in Buenos Aires, it seems impossible that they could fund a move for a player who cost City £25million.
Chelsea and Inter Milan are among other potential suitors for the former West Ham player, whose relationship with Eastlands boss Roberto Mancini is said to be at breaking point.
Wanted: Wolfsburg's Dzeko
In Madrid, there are conflicting reports about how long Higuain will be sidelined by a back injury. He has been Mourinho's clear first choice as the spearhead of a three-man attack and if Benzema is out, the boss would most likely have to play Cristiano Ronaldo in a central role.
The club have been looking for a physically impossing No 9 rather than someone of Tevez's stature, though getting rivals to part with players in January is proving difficult.
They like Athletic Bilbao's Fernando Llorente, Wolfsburg's Edin Dzeko and Anderlecht's Romelu Lukaku - with City's unsettled Emmanuel Adebayor also under consideration.
Despite Tevez being just 5ft 8in, however, he plays as a lone striker for City and has previously spoken about being comfortable as a No 9 or more creative No 10.
His employers are likely to step up their own bid for Bosnia and Herzegovina hot shot Dzeko, though they insist they will fight to keep hold of their captain.
'It is with disappointment that we confirm to our supporters that Carlos Tevez has submitted a written transfer request,' they said in a statement. 'The club can also confirm that the request has been rejected.
'Carlos' current five-year contract has three-and-a-half years to run and he is the highest paid player at the Manchester City Football Club.'
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