Conn-Clarke has enjoyed a hugely successful season in the National League with Altrincham, but the Greater Manchester side failing to get promoted means he is likely to leave.
Ipswich, now in the Premier League, have been credited with interest in the attacker, while Peterborough have already seen an offer turned down.
Posh still want Conn-Clarke and are acting early before the season has even ended, but they are facing competition from Wigan.
Wigan are looking to spend cash to strengthen this summer as they eye pushing for promotion from League One next term.
Peterborough are renowned for their eye for non-league talent and believe Conn-Clarke could be their next big thing.
A £250,000 fee has been mooted as possible for the Altrincham man.
It remains to be seen if Premier League new boys Ipswich will go in with an offer, while Portsmouth have also been linked with Conn-Clarke.
