McArthur has claimed that Scottish Premiership giants Celtic are interested in signing the 17-year-old forward in the summer transfer window.
The talented youngster is out of contract at Dunfermline, who have offered him a new deal.
“We are currently going back and forward (with Celtic) with regards PJ; we are in the process of negotiations,” McArthur told Deadline News.
“We’ve put our cards on the table and we’ll see if we can reach an agreement. He is out of contract, so is free to speak to other clubs, but in his case there’s compensation due.
“But we’ve made him a contract offer, because we would like to see him stay at Dunfermline, and that is still on the table.”
Crossan featured mostly for Dunfermline’s Under-20 side this past season and made his breakthrough into the first team towards the end of the 2015-16 campaign.
The Scottish youngster featured in Dunfermline’s wins against Stranraer and Brechin City last season.
Due to Crossan’s age and relative lack of experience, he is unlikely to be part of Celtic’s first-team plans next season.
The youngster is very talented and hugely promising, and the Hoops probably see him as part of their long-term plans.
If Celtic do sign Crossan in the summer transfer window, then perhaps they could loan him out to a team in the Scottish Championship or perhaps even in the Scottish Premiership in order for the youngster to continue with his development.