The visitors, meanwhile, have the chance to clinch back-to-back victories after beating Norwich City 2-1 at the weekend.
Match preview
Kompany has experienced mixed results in his opening three games in charge of Burnley, with their four-point tally leaving them in 11th place ahead of the midweek contest.
The season got off to a promising start with a 1-0 away win over Huddersfield, while they had to settle for a point on matchday two following a 1-1 draw against Luton Town at Turf Moor.
After beginning his tenure with four points from two games, Kompany suffered his first defeat as Burnley boss, losing 1-0 to Watford, who were relegated from the Premier League alongside the Clarets last season.
A Tom Cleverley strike in first half stoppage time proved enough to condemn Burnley to defeat, but they have the chance to offer an immediate response on Tuesday.
Kompany will hope his team can claim their first home victory of the season and continue to remain solid defensively after conceding just two goals in their opening three games.
Meanwhile, Hull will try to maintain their unbeaten start to the Championship season when they travel to Turf Moor on Tuesday.
The Tigers began the campaign with a 2-1 win over Bristol City before being held to a goalless away draw against Preston North End.
Despite losing to Bradford City in the first-round of the EFL Cup, Hull returned to winning ways at the weekend when they played host to Norwich.
Summer signing Oscar Estupinan opened his account for the Tigers, scoring a brace that helped his side claim a 2-1 victory against the Canaries.
As well as maintaining their unbeaten league start, the second-placed side will attempt to clinch their first away victory of the season on Tuesday.
Burnley Championship form:
Hull City Championship form:
Hull City form (all competitions):
Team News
The hosts will be without the services of Ashley Westwood and Johann Berg Gudmundsson, who are sidelined with ankle and leg injuries respectively.
Jack Cork was introduced as half time substitute against Watford, and the midfielder could be given a starting place for Tuesday's contest.
After making his debut with a second half appearance at the weekend, Nathan Tella is in contention to start for the first time in a Burnley shirt.
As for Hull, they are still unable to call upon Brandon Fleming, Dogukan Sinik and Nathan Baxter, who are all absent due to injury issues.
Following a 2-1 win over Norwich at the weekend, Shota Arveladze could be tempted to name an unchanged lineup for the midweek encounter.
Estupinan will continue to lead the line for the visitors, with the 25-year-old looking to add to the two goals he scored at the weekend.
Burnley possible starting lineup:
Muric; Roberts, Harwood-Bellis, Taylor, Maatsen; Tella, Bastien, Cork, Brownhill, Cullen; Barnes
Hull City possible starting lineup:
Ingram; Coyle, Figueiredo, Greaves, Elder; Jones, Slater; Sayyadmanesh, Tufan, Tetteh; Estupinan