Anigo took charge for the Ligue 1 club's 2-1 UEFA Champions League defeat to Borussia Dortmund on Wednesday following the sacking of Elie Baup.
The 52-year-old, who normally serves as Marseile's sporting director, has coached the Provence club on an interim basis on a number of occasions.
And, while Anigo insists he is only focusing on Marseille's final three games before the mid-season break, the former defender is open to the idea of extending his spell as coach.
"I'm not quick to speculate about my own future - my mission is four matches, and three remain," he said.
"When doing this job, in this club, you cannot help but be affected: I've worked for this club for ten years, it gets better but it is also sobering.
"I'm not in thinking of my project beyond January. I am not against continuing, if the president cannot find anyone who suits him.
"But once everything has been said the most important match happens against Lyon on Sunday."
Marseille sit fifth in Ligue 1 - 13 points behind leaders Paris Saint-Germain - and have won just one of their last five games in all competitions going into their trip to Lyon.
‘Maybe I’ll be signed by Wrexham, who knows? English football would suit me – if an offer came, I wouldn’t think twice, I’d go immediately’ Euro 2024 cult hero reveals his UK dream
‘Managing Leeds? It was an option that appeared, but it wasn’t the right timing. I decided it wasn’t a good idea to leave the club I was at mid-season’: Premier League boss admits to turning down opportunity to replace Jesse Marsch in 2023
