Koeman is facing increasing pressure after Benfica comfortably beat his Barcelona side in the Champions League, with Belgium head coach Martinez the favourite to replace him
Koeman has endured a disastrous spell in charge of the La Liga giants since taking over last year, with Barca currently sitting sixth in the Spanish top-flight.
While that is far below what is expected at the Nou Camp, things have gone even worse in the Champions League, with Koeman’s men currently bottom of Group E.
They are still yet to score in the competition this season, having been dominated by Bayern Munich and humiliated by Benfica in their opening two games.
The lowest point of the season so far came in Portugal on Wednesday, with Barcelona having just a single shot on target as Benfica comfortably won 3-0.
The defeat also ensured that Barca began their European campaign with two losses for the first time since the 1972/73 UEFA Cup.
Only Louis Van Gaal has a worse record in the Champions League group stages as Barca boss than Koeman.
And with the latter entering into a war of words with club president Joan Laporta, his reign appears to be hanging by a thread.
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Belief that the end is near for Koeman at the Nou Camp increased in the aftermath of their desperately poor visit to Portugal.
Koeman even hinted that he does not believe that he has the support of the board in his post-match press conference.
“I feel supported, very supported by my players here at Barcelona. And by the club? I don’t know,” he said.
“I’m not going to argue about the level of my team. There’s no point in comparing this with Barcelona teams of years gone by. That’s as clear as water.
“I can only give you my opinion of my work at the club – I feel backed by my players and their attitude.”
The man many believe will eventually replace Koeman in charge of Barcelona is Martinez, who has enjoyed a relatively successful spell with the Belgian national team.
And now the Spaniard has dropped a cryptic hint that he could return to his home country to replace Koeman.
“There is absolutely nothing – and there has been no contact with Barcelona. I want to fulfil my contract, but many things may arise along the way,” he told HLN.
Though his contract runs until after the 2022 World Cup in Qatar, his insistence that “many things may arise” could be seen as an attempt to open the door to a move to the Catalan club.
He led them to the semi-finals of the 2018 World Cup, though they were knocked out at the quarter-final stage of Euro 2020.
Still, Martinez has proven he has the ability to work with world-class players having worked with Kevin de Bruyne and Romelu Lukaku with Belgium.
He also has experience of winning major trophies and operating under financial restrictions, having won an FA Cup with Wigan, who had one of the smallest budgets in the Premier League.