The Everton defender could line-up against his teenage son Tyler in next month's third round.
Tyler plays for League One's Peterborough, who have been drawn away to the Toffees.
The two sides will meet at Goodison Park on January 9, with the Everton squad relishing the chance to face their team-mate's boy.
Asked what the players have been saying to Young about facing the 18-year-old, Tarkowski told talkSPORT Breakfast: "There's been a few lads asking what position he plays in so they can kick him.
"But there's no-one that wants to kick him more than Ash himself.
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"So I think Ash is hoping he plays."
Tyler joined Posh from QPR in the summer and has since made his first-team debut in the EFL Trophy.
The young midfielder has also been named on the bench in both the league and the Carabao Cup already this term.
He could now come up against his former England international dad next month.
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Young has been an important figure for the Toffees this season, despite turning 39 during the summer.
Tarkowski added: "Ash is incredible. You wouldn't think he's the age he is.
"You played with him yourself years ago. The level he's at still, physically. Even just to stick at it day in, day out in training.
"It's not easy at times. I've been through it myself a little bit. But he's a true professional. At it all the time.
"And he's incredible around the press as well. We love him here."
The winger-turned-full-back is into his second season at Goodison Park after extending his contract back in June.
So far this season, Young has made 12 Premier League appearances to take his overall tally in the competition to 465.
He has also scored once and provided three assists for his team-mates.
His free-kick against Wolves last week means he is now just one goal away from 50 Premier League goals.
Meanwhile, the strike also made him the oldest player to score a free-kick in the competition as well as the fourth oldest scorer overall.