Eighth-placed Boro are currently three points behind the top six, while the Robins' horrific recent run of form has left them in 15th position.
A win for Middlesbrough on Tuesday would draw them level on points with Bournemouth in sixth before the Cherries face a crucial clash against seventh place Cardiff City 24 hours later; Boro themselves then play the Bluebirds at the weekend.
Neil Warnock's side made a statement of intent on Saturday when they won 2-0 away from home at fifth-placed Reading, with both goals coming in the space of seven first-half minutes from Ashley Fletcher and Marc Bola.
Warnock said afterwards that "it was as good a performance against a good side since I have been here", and the result marked a second victory in a row for Boro after they had been winless in their previous five.
While the win brought just their second clean sheet in 12 league matches, only five Championship clubs have conceded fewer goals than the Teeside outfit this season.
Middlesbrough's 2-1 victory over Huddersfield Town in their last home game ended a run of four successive defeats at the Riverside.
George Saville's goal earned Boro a 1-0 win against Bristol City in this season's reverse fixture at Ashton Gate – only the third time that they had triumphed over the Robins in their last 12 attempts.
The home side may well feel confident of another three points on Tuesday, though, as Bristol City appear to be a club in crisis on the pitch.
A 1-0 defeat to Barnsley on Saturday was their seventh successive loss in all competitions, a run which has led to them dropping from just outside the playoffs to mid-table and 10 points behind the top six.
Since the start of 2021, only Huddersfield and Wycombe Wanderers have picked up fewer points than Bristol City and only the Chairboys have conceded more goals.
Having won five of their first eight away league games of the season, the Robins have now lost their last seven on the road in the Championship, as well as exiting the FA Cup away to Sheffield United.
The run of fixtures does not get any easier for them either as after their trip to the Riverside, they face top-six opposition in the shape of Swansea City and Bournemouth.
Middlesbrough Championship form: LDLLWW
Bristol City Championship form: LLLLLL
Bristol City form (all competitions): LLLLLL
Warnock has switched to a back three for Middlesbrough's last two games and won both of them so he is expected to stick with the same system on Tuesday night.
Simpson admitted that Bristol City had 15 players unavailable against Barnsley and that none would be ready to return at the Riverside.
The injury list could get even longer as Henri Lansbury was taken off on Saturday with an Achilles issue.
Steven Sessegnon will be assessed ahead of the trip to Teeside after he played 76 minutes against Reading on his return from a lengthy hamstring injury, while Nahki Wells could come back into the side upfront.
Middlesbrough possible starting lineup:
Bettinelli; Dijksteel, Fry, McNair; Fisher, Morsy, Howson, Saville, Bola; Fletcher, Watmore
Bristol City possible starting lineup:
Bentley; Moore, Kalas, Mariappa; Hunt, Vyner, Massengo, Palmer, Sessegnon; Diedhiou, Wells
We say: Middlesbrough 2-0 Bristol City
It is difficult to see past a Middlesbrough win in this one; Boro seem rejuvenated since Warnock's switch to a back three and their win at Reading will give them a lot of confidence. Meanwhile, Bristol City are at the other end of the spectrum – rock bottom with little hope of picking up three points.