The Hoops full-back has been linked with a £5million switch to the Barclays Premier League with Norwich City.
It is not the first time, nor is it likely to be the last, that the Honduran ace has been tipped to move down south.
But Didier Agathe, who enjoyed many European nights during five years in Glasgow, reckons the left-back would be crazy to leave now.
For Neil Lennon's team is on the verge of securing a place alongside the cream of the continent's clubs in the Champions League group stage.
After defeating HJK Helsinki 4-1 over two legs in the third qualifying round, the Scottish champions now face Swedish side Helsingborgs in a play-off to qualify for the first time since 2007.
With teams such as Bayern Munich, Chelsea and Real Madrid possibly awaiting them if they do make it, Izaguirre should stay exactly where he is for the time, being according to Agathe.
"I think it is only when you play in a big European night at Parkhead that you really appreciate how big a club Celtic is," he explained.
"I played for Celtic against some of the biggest clubs in the world when I was in Glasgow. But we always did well against them, especially at home.
"We beat Porto, Rosenborg and Juventus at home in our first season in the Champions League group stage. We also beat Lyon and Anderlecht in that competition and drew with AC Milan and Bayern Munich.
"Then, of course, in the Uefa Cup, we knocked out Barcelona. These clubs may have more money to spend on players, but the Celtic support is as big and passionate as any in the world. The atmosphere on European nights is incredible.
"I think Emilio will love being part of the Champions League if Celtic get there. It is too big an opportunity for him to turn down. Plus, if he does well in the Champions League, who knows which clubs could be interested in him?"
Izaguirre won a clean sweep of Player of the Year awards – taking the SPL, SPFA and SFWA gongs – after a stunning debut campaign in Scotland in the 2010/11 season.
However, the 26-year-old was out for most of last season after suffering a broken ankle in a league game against Aberdeen at Pittodrie in August.
He is, though, fully fit now, and returning to his old self.
Agathe, now running a soccer academy in his native Reunion Islands and hoping to supply his former club with up-and-coming talents, is hopeful that Izaguirre and his team-mates have what it takes to emulate the feats of the team he played in back in 2001.
Martin O'Neill's star-studded side reached the group stage for the first time in their history that season following a stunning win over Dutch giants Ajax.
Agathe scored along with Bobby Petta and Chris Sutton in an amazing 3-1 win in the first leg in the Amsterdam Arena and they went through after losing the rematch 1-0.
Until the impressive wins over Helsinki, Lennon's record in qualifying matches in Europe left a lot to be desired.
His teams had lost to Braga, Utrecht and Sion, and only got into the Europa League last term after their Swiss rivals were expelled for fielding ineligible players.
"The Celtic players have not been involved in the Champions League, but the manager and his assistant Johan Mjallby have," said Agathe.
"Five years is a long time for a club like Celtic to be out of the Champions League and I know Neil and Johan will be desperate to get the club back to that level this season.
"The players have done very well to win the league, but at a club like Celtic it is not acceptable just to do that. They must go on and qualify for the Champions League as well.
"It is the stage the club and the players should be involved on. But Neil, like Martin O'Neill before him, has a good team will know how to prepare his players for these matches."