A loss shouldn't necessarily derail City, which is 1-50 with some British bookmakers to win the league.
Shakhtar insist it is below their valuation - but the move is a signal of City's determination to land their man.
Whereas a run to semi-finals would have been considered a notable success in those days, the is now some way down the list of priorities, even at this late stage of the tournament.
"They were really good, so we tried so hard, all the players tried and we're really happy", added the 46-year-old, who has lost five of his last six semi-finals including City's last domestic loss to Arsenal in last season's FA Cup. So most of the intrigue for the remainder of the season could lie in the fight for Champions League qualification behind City.
Jupp Heynckes has led Bayern to 15 wins from 16 games across all competitions since returning as coach, the only blemish being a loss at Borussia Moenchengladbach.
Manchester City, nevertheless, should have scored the opener and broke Bristol City's resolve.
Barcelona, coming off four straight wins in the league, visits mid-table Real Sociedad on Sunday.
However, City boss Pep Guardiola has moved quickly, with the Manchester Evening News reporting they have tabled an opening offer. Kevin De Bruyne did better, forcing Fielding into the best save of the first half with a curling shot from the left flank.
The visitors, though, were not in the least intimidated and went close themselves immediately afterwards, Claudio Bravo saving Joe Bryan's long-range shot.
It is understood that Mourinho is also chasing Valencia's Carlos Soler and Atletico Madrid's Saul Niguez, but the latter's release clause of 140 million might deter further advances.
The 32-year-old has played a key role for City this season, but it has been suggested that Guardiola is wary over the amount of games that the central midfielder has competed in since August.
"He was in Barcelona this week and the news is quite good".
Aden Flint ensured the Championship side went into the break 1-0 up when he produced a superb header to clear over his own bar as Raheem Sterling's shot looked destined to bring the sides level.
The lead lasted until the 55th minute when De Bruyne drove forward and fed Sterling, who pulled the ball back for the Belgian to lash home from 15 metres.
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