
After four months inside the top six, City were dislodged by Phil Parkinson, their former manager, on Tuesday night, after Tigers drew 2-2 at relegated Leicester City, and their rivals beat Oxford United 1-0 at the Kassam Stadium.
Both clubs are locked on 70 points, four ahead of Derby County, but Wrexham's goal difference is just two better than the Tigers, with all to play for over the weekend.
JOIN US ON FACEBOOK! Latest Hull City news and analysis on our Hull Live FB page
City play first with a trip to Charlton Athletic, who are not yet safe from the drop and need a point to rubber-stamp their survival, before Wrexham go to champions Coventry City on Sunday lunchtime.
Both clubs then return to their home grounds for final day meetings against Middlesbrough and Norwich City respectively, with all to play for.
Dawson, though, believes the Tigers will need to collect maximum points to regain their place in the top six, one they had held for 137 days.
"They're two goals behind Wrexham, they got a good win (at Oxford)," Dawson said on Sky Sports. "What have they got - Coventry and Middlesbrough in the last two games? So Middlesbrough are certainly going to have something to play for come the final day, a lot of football still to be played.
"Hull (City) can only affect now (their own results). They've had no wins in five, so they've put themselves in this position. They've been in the play-offs for so, so long, and then when you're getting chased, and you get caught. Two games to go, still right in the mix, but they've got to start winning football matches. Two to go, they've got to win both in my opinion."
Another former City ace, Curtis Davies, believes playing first this weekend could be a benefit for the Tigers, providing they can secure a first win since beating Sheffield Wednesday last month.
"Wrexham don't play until Sunday, so it's if they can lay a marker, that will be a really interesting game on Sunday to see how Wrexham react," he added.
Here at Hull Live, we are dedicated to bringing you the best Hull City coverage and analysis. Make sure you don't miss out on the latest City news by joining our free WhatsApp group.