Despite having won the competition twice, with Porto and Inter Milan, the Portugese is under increasing pressure now that the Premier League champions have been beaten five times in their past seven games.
They have slumped to 15th place in the table and key players such as last season's player of the year, Eden Hazard, are badly out of form.
Although fans remain generally supportive, chanting Mourinho's name during the 3-1 home defeat by Liverpool on Saturday, some players are becoming disenchanted, according to media reports.
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That means it is another big week for him, with the Kiev game followed by a visit to Stoke City, who knocked Chelsea out of the Capital One Cup on penalties last Tuesday.
The goalless draw away to Dynamo two weeks ago was one of Chelsea's better performances of a poor season. They hit the post twice and could easily have won, but failing to do so left them on four points in Group G, one behind the Ukrainian champions.
Porto top the group with seven points and could extend their lead against Maccabi Tel Aviv, who have lost every game so far.
"Everyone can see how much hard work we are putting in and we are playing well, but the results are not coming for us," Nigerian midfielder Jon Obi Mikel told reporters after the defeat by Liverpool.
"The goals and the wins are not coming, but you can see we are playing for the manager and we are right behind him. There is no doubt about it. If anyone can turn this around I definitely think he is the one."
Kiev's Portuguese international Miguel Veloso also backed his compatriot Mourinho, with a post on his Facebook page.
"Say whatever you want to say," he wrote.
"Unfortunately, I had the chance to exchange a few words with this man only twice in my life. I already used to be his fan, now I am even more."
Dynamo, who have no injury problems, won 2-0 against Metalist Kharkiv at the weekend, a fourth successive win that kept them second on goal difference behind Shakhtar Donetsk at the top of the table.
In the Champions League, they scored a late equaliser to draw 2-2 at home to Porto and then won 2-0 in Tel Aviv before the first Chelsea game.
(Writing by Steve Tongue, editing by Stephen Wood)
