The Honduran defender, who turns 28 on Saturday, is about to enter the final year of his contract at Parkhead.
That could alert clubs in England, especially as Izzy has now recaptured the form which made him a signing target in his first season with the Hoops.
Izaguirre is now fully recovered from the broken ankle which he sustained at the start of his second season in green and white.
And, with another appearance in the World Cup finals on the horizon - it was the stage where he first caught the eye of Parkhead boss Neil Lennon when he played for Honduras against Spain at the 2010 finals in South Africa - Izzy could again be in the spotlight when he lines up in Brazil against France, Ecuador and Switzerland.
Izaguirre yesterday became, alongside Rangers' Arnold Peralta, the first Scottish-based players to be given a ticket to Brazil.
And he will once again be one of the key men for the unfancied nation, ranked 32 in the world.
"I could not please everyone with this selection, but I have tried my best," boss Luis Fernando Suarez said.
"I have great faith and trust in these players and believe they will do Honduras proud at the World Cup."
A World Cup on Brazilian soil will be seen as the ultimate football experience for some players.
It will be Izaguirre's second participation at the finals, but the Hoops hero has no intentions of packing his bags again on his return to Glasgow later this year.
Izaguirre said: "I appreciate all the things Celtic have done for me. I have one more year (of my contract) left, but I will sign here no problem."
Izzy joins Mikael Lustig and Kris Commons in entering the final year of his current deal.
But, like his two team-mates, he recognises he is already at a club where he is winning trophies and developing as a player.
"I am happy here as Celtic opened the door for my career in Europe," explained the man who signed from Honduran club Motagua for just £640,000 in 2010 after a number of clubs down south passed on the chance to recruit him.
Izaguirre has had no regrets about that, and said: "I want to play in the Champions League every year and win more leagues and cups.
"I have enjoyed it here and I am very happy, but I want more joy next season."
Izaguirre suffered bitter disappointment at the end of his first season at Celtic when they just lost out to Rangers in the title run-in.
However, he has picked up championship medals in each of the three years since then, and aims to continue being a serial winner.
Izaguirre said: "When I signed for Celtic, I thought it was a club that won titles every year, and I am happy that I have won two cups and three leagues. Next season, I will try to win all of the trophies."
Izaguirre said his goodbye to Parkhead - but only for this season - at the weekend as he bowed out in style by helping the Hoops to a 5-2 win over second-placed Aberdeen.
He has being given time off by boss Lennon to recharge his batteries ahead of the World Cup and will miss the meeting with St Johnstone tonight and Sunday's Parkhead party against Dundee United.
Another winner's medal has only added to the left-back's sense of enjoyment at life in Scotland.
And it is a feeling shared by his family, especially son Emilio Jnr. Izzy said: "It's a really good moment for me to see my family so happy.
"Emilio loves football. He plays every day and supports Celtic. He wants to play all of the time, but sometimes I am tired."
In Izaguirre's absence, Celtic will head to McDiarmid Park tonight aiming to continue their charge towards the 100 points barrier in the Premiership.
It would be another notable achievement for Lennon's side, who extended their lead at the top of the table to a huge 30 points with an eye-catching triumph over the Dons.
"We are hungry to win every game," Izaguirre told the Celtic matchday programme. "We try every week to win and keep clean sheets.
"It's very hard when we play against the other teams. We need to be stronger than they are and score quickly.
"It can be very difficult playing away from home, but we have done very well this year.
"There is a strong togetherness in the squad, and we have been playing better and more consistently in the league.
"This is my fourth year here, and we all play for the team. It isn't possible to win every game or play the same way every week. But, we try to do that."
Izzy's return to top form has been a major factor in Celtic's cruise to the title this season.
But he is already setting his sights on even more glory next time around.
He said: "The target at the start of the season is qualifying for the Champions League. After that, we concentrate on trying to win every game in the league.
"We didn't do well in the cups, but we will try to do better in those next year. It's special to win a title as you work hard for a whole year, and it's nice to get the trophy. But, what is important is doing well again next season."