Crystal Palace are yet to issue an update on the 76-year-old's health since confirming he was 'stable and undergoing tests' in hospital after falling ill at the club's training ground.
Reports that Eintracht Frankfurt boss Oliver Glasner would imminently replace Hodgson at relegation-threatened Palace had ramped up prior to the ex-England boss' illness, and the 49-year-old Austrian was spotted at upcoming Eagles opponent Tottenham's 2-1 Premier League loss to Wolves on Saturday.
Writing in a column for Mail on Sunday, Murphy said: "I messaged (Roy) after the worrying news that he'd been taken ill at Crystal Palace's training ground and was reassured by his replies. He said he was feeling fine, taking stock and prioritising his health.
"It was good to hear. At 76, he's absolutely right to put himself first - and he indicated he's starting to realise that himself.
"Roy has dedicated his life to football and been incredibly successful. He's won trophies, travelled the world and given so much to so many players - including myself when we reached the Europa League final with Fulham.
READ MORE ON ROY HODGSON
Crystal Palace's meeting with Roy Hodgson's replacement gatecrashed by Championship stars
Crystal Palace agree Oliver Glasner deal with Roy Hodgson set to leave
"He can be very proud of what he's achieved and has nothing to prove. I'm sure he felt an enormous sense of responsibility in a tough campaign for Palace, but if this is a full stop on his managerial career, he should step away with full peace of mind."
talkSPORT understands Palace have agreed a deal for Glasner to replace Hodgson that would secure the 2022 Europa League-winning manager's services through to 2026, with the option of a further year.
Palace were unable to provide an update when contacted by the PA news agency.
Hodgson had been reluctant to use the word 'retire' after stepping away from both his first spell in charge of Palace and his stint at Watford during the second half of the 2021-22 season, but in recent weeks seemed more definitive about his future.
Most read in Football
Newcastle place Ashworth on gardening leave as he tells them he wants Man United move
Bayern CEO gives two-word response to question on Tuchel future after damaging defeat
De Zerbi and Sheffield United player give surprising takes on Holgate tackle
Ex-Arsenal players are putting Harry Kane's trophy chances in jeopardy
At one point, he declared: "The fact is that this is my last spell in football" before admitting he might be tempted if offered something "within the world of football that actually suits me."
The sides separated by just five points, though the likeliest candidates are Hodgson's right-hand man Ray Lewington and assistant manager Paddy McCarthy.
The pair took charge in September, when a previous illness landed Hodgson in hospital, forcing him to listen to Palace's defeat to Aston Villa on the radio after technical issues locked him out of a live stream.
He told reporters in the following press conference: "We were sent the link but we couldn't get it to work, unfortunately, so we had the even worse experience of listening to it on the radio and that's not something I would recommend."