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The Republic of Ireland midfielder hasn’t featured since last December after suffering an ACL injury, which required surgery.
Cullen is set to play 45 minutes for the Under-21s against West Ham tonight and has a chance of making the squad at Lincoln if all goes to plan.
However, Jacob Bruun Larsen is facing 10 to 12 weeks out with a stress facture and Jamie Vardy will miss Saturday’s game due to a back issue.
“(Cullen) is really important and you only see it already on the training field,” said Hayen. “When Josh is there, he understands the game, he reads the game, talks to the players.
“You see a big difference - when do we press, the right moment to press. If someone is alone he says he has to wait and then step together, so it's a big impact.
“Now we're only talking about Josh but we have nine injuries. If you see how many injuries we’ve already this season with Bash (Humphreys), Anel (Ahmedhodzic), Josh, Hannibal, Aaron Ramsey, now Jacob has a stress fracture for 12 weeks, Jamie is out.
“Those are nine players that if you have them in the selection, you have a lot of solutions but for the moment, they're not available and that's really unfortunate. But that's the situation.
“I didn't mention Zeki (Amdouni). He had a really good international break, comes back, scores against Watford, is out. Then Jamie comes into the team, he scores, he's out so this is actually a little bit the story of all season.”
The Clarets are aiming to bounce back from last weekend’s defeat against Wolves as they head to Lincoln, who earned a point at Millwall last time out.
The Imps, who won League One last term, currently sit 18th in the table with 12 points from their opening 11 games.
Hayen knows it will not be easy at Sincil Bank and insists Burnley must be prepared for Lincoln’s physical threats.
“They play quite direct and have a lot of physicality with set pieces, with throw-ins, with free kicks, but also their direct style of play,” said the head coach.
“We have to be aware for that but still there will be possibilities. We have to deliver the same quality like we did in the first 35 minutes (against Wolves) because if we do this, we can hurt everyone.
“But if we fall back to an average level like we did the last 60 minutes, then it becomes difficult. But we have started the week positive and this is the mindset we want to show Saturday.
“We have to match the duels and the second balls, but we don't have to play the same game like they are doing because this is not the way we want to play.
“We want to play in a good way, of course, much more intelligent, using our moments and if we are in lead, keep that lead.”