The 23-year-old has been given chances by Iraola during preseason, but he was left out of the Reds matchday squad for their Premier League opener, as Dominik Szoboszlai's late penalty snatched a 2-2 draw at Newcastle United.
The England international spent last season out on loan at Aston Villa, but Unai Emery opted against a permanent deal, with the former Fulham star effectively frozen out to avoid triggering a £35M purchase clause in the second half of the campaign.
Elliott is now into the final year of his contract at Anfield, and talkSPORT pundits Jeff Stelling and Ally McCoist have called on Newcastle to make a late offer for him before the window closes on September 1st.
"His contract is up next summer, so he's not going to cost a King’s ransom. What are we talking, £15-20million? Something like that," Stelling stated.
"I was going to look at the newly promoted sides like Coventry, but would he be out of place at Newcastle?"
"He wouldn’t look out of place in Newcastle's team and they're looking for players who are similar to the ones they’ve lost. Obviously, Anthony Gordon springs to mind," agreed former Scotland star McCoist.
"The most important thing, and it’s massively important for Harvey Elliott this season, last season was a write-off, he can’t afford another one like that."