The Welsh side have won both of their group matches and will host the winless Russian outfit, who recently sacked their manager, as they aim to edge closer to the knockout stages
The Premier League side opened the competition with a memorable 3-0 victory in Valencia, before battling to a 1-0 win over Swiss side St Gallen at the Liberty a fortnight later.
Michael Laudrup's men have scored four goals without reply during the group stage - which is sure to cause concern for winless Kuban.
In contrast to Swansea, Kuban - who replaced manager Dorinel Munteanu with BATE Borisov boss Viktor Goncharenko last week - have conceded four without finding the net, losing 2-0 in Switzerland and at home to Valencia.
Swansea recovered from back-to-back league defeats to Arsenal and Southampton by putting Sunderland to the sword in a 4-0 victory on Saturday and currently sit 11th in the Premier League.
Kuban, meanwhile, are having a torrid time in the Russian Premier League and have picked up just one win from their last nine fixtures in all competitions - Goncharenko's first game in charge ending in a 3-1 defeat at Dynamo Moscow.
They also lie in 11th, and are already nine points adrift of the European qualifying positions after 13 matches.
However, Goncharenko has no qualms about the magnitude of the task he has taken on at Kuban.
"There is no such thing as an impossible task," the 36-year-old told the official Uefa website.
"If you want to talk about the big tasks that we are working towards, then we want to play attractive football and win. I think Kuban have good players and we can do what we have set out to do."
Laudrup will be forced to cope without Ben Davies, who picked up an ankle injury in the win over Sunderland, while captain Ashley Williams (ankle) remains a doubt having been absent from first-team and Welsh international duties since Swansea's 2-0 victory at Crystal Palace on September 22.
Winger Pablo Hernandez had been expected to be involved at the weekend, but may make his return from injury on Thursday instead.
Alexei Kozlov and Charles Kabore were missing for Kuban in the match against Dynamo Moscow on Sunday, and remain doubts for the trip to south Wales.