The 24-year-old has been on the fringes of the squad this season, making only eight appearances across all competitions.
The Hammers were reportedly willing to listen to offers for the young full-back, whose contract is set to run out in the summer.
Palace explored whether the Hammers would consider a sale before the transfer deadline.
However, The Athletic’s David Ornstein says their enquiry has been knocked back by West Ham, who are planning to offer him a new contract.
Spending on Premier League transfers has been down this month due to Financial Fair Play restrictions, but activity has picked up in recent days.
Palace have completed the signing of Daniel Munoz from Genk, while Blackburn Rovers midfielder Adam Wharton is also set to arrive at Selhurst Park.
The Eagles also wanted to add Johnson, but it appears he will stay with the Hammers at least until the end of the season.
The right-back is an academy product of West Ham, who made his competitive debut in February 2019 against Manchester City.
He has made over 90 senior appearances for the English club but has never nailed down a place in the first team.
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