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Johnstone is a free agent this summer after turning down the chance to sign a new contract at Championship club West Brom.
The goalkeeper has been the subject of plenty of interest, but he is now closing in on a move to Palace.
Standard Sport reported Palace’s interest in Johnstone earlier this week and it is understood talks have taken place since then.
The 29-year-old is excited by the prospect of moving to Selhurst Park and the Eagles are now the favourites to sign him.
Johnstone, who has three England caps to his name, is keen to play in the Premier League after spending last year in the Championship.
He had been tracked by Tottenham earlier this summer, as Spurs want to sign a homegrown goalkeeper to be back-up to Hugo Lloris, but they swooped for Fraser Forster instead after he left Southampton for free.
Johnstone’s arrival would mean Palace have three goalkeepers competing to be No1, with Jack Butland and Vicente Guaita already on their books.
One of Butland or Guaita would be expected to leave if Johnstone came in to make room for him in the squad.
Butland was Palace’s No2 last season, but he performed well in their run to the FA Cup semi-finals. Guaita, meanwhile, is 35 and has previously spoken about his desire to again play for Valencia before his career is over.