As revealed by talkSPORT's Saudi correspondent Ben Jacobs, the Magpies have reached an agreement with Al Shabab over the Paraguayan, who is open to the move.
The deal is not done yet, though, but should it go through it would be a huge blow to Newcastle fans, with Almiron a big favourite at St James' Park since his 2019 arrival from Atlanta United.
The winger wrote himself into the club's folklore in October when he scored their first Champions League goal in two decades.
Almiron opened the scoring in a stunning 4-1 win over Paris Saint-Germain, lifting the roof off St. James' Park.
However, since then things haven't gone brilliantly for Eddie Howe's side, finishing fourth in their Champions League group, and also dropping to mid-table in the Premier League.
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The downturn in form has been coupled with concerns over Newcastle's ability to comply with financial fair play.
The Magpies had a mammoth net spend following their takeover by Saudi Arabia's Public Investment Fund in October 2021, and have now admitted they may have to sell to spend again.
The sale of Almiron could help with that, with Al Shabab not among the four SPL sides also owned by the PIF.
Al Shabab are looking for a new poster boy after allowing Ever Banega to return to Argentina, and they've also approached the recently sacked Jose Mourinho, according to reports.
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Magpies CEO Darren Eales had earlier revealed the club may have to sell to buy, and Howe later confirmed that to talkSPORT.
"I don't think there's any surprise in that because every football club will have to trade now with Financial Fair Play being such a prominent part of football," he said.
"So selling players and trading is part of what every football club is going to have to do and we're no different.
"So whether it's our top players as you quote, that remains to be seen but selling players and trading will be part of football going forward now I think."