The Tigers have secured just one victory in their last six league games, a 6-0 win over Fulham, while Aitor Karanka's side have won three of their last four matches
Steve Bruce's charges have prospered after returning to the Premier League, collecting 23 points from 20 matches to sit 10th in the table.
However, Hull can expect a stern challenge from a Middlesbrough side who have looked revitalised in the last month under Aitor Karanka.
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Middlesbrough head into this weekend's game unbeaten in four league games, having won three in a row prior to surrendering a two-goal lead in a 2-2 draw with Bolton Wanderers on New Year's Day.
Although Karanka's men sit 15th in the Championship, they are only six points adrift of the play-off positions.
Neither Middlesbrough or Hull have won the FA Cup, although the former reached the final in 1997 before suffering a 2-0 defeat to Chelsea.
Hull make the journey to Middlesbrough having won just one of their last seven league matches.
Yet that victory proved a hugely convincing one as Fulham were thumped 6-0 at the KC Stadium on December 28.
Bruce was keen to rest players for the 2-0 defeat at Liverpool on New Year's Day, which capped a hectic end-of-year schedule, but acknowledges he simply does not have the necessary depth in his squad.
"You ask yourself whether you could have freshened it up and made five or six changes but we've not got that luxury yet," he said.
"We can't afford to do it. Overall, the squad and their attitude is great."
Middlesbrough and Hull last met in the FA Cup seven years ago, also at the third-round stage.
After a 1-1 draw at the KC Stadium, Middlesbrough progressed courtesy of a 4-3 triumph in the resulting replay.