Former Netherlands international Koeman last month revealed his decision to leave the Eredivisie club at the end of the season, bringing an end to his three-year tenure in Rotterdam.
Rutten - who has spells in charge at PSV, Schalke, Vitesse and Twente on his CV - will pen an initial one-year deal at De Kuip.
The 51-year-old has paid tribute to Koeman, who continues in the dugout until the end of 2013-14, and is excited by his new challenge.
"The club has obviously done fantastically under Ronald Koeman in recent years," he told the club's official website.
"After some lean years they have built something up in Rotterdam, from both a financial and sporting perspective.
"The broadly shared ambition to continue on the successful path can be seen and felt in everything. I would like to contribute to that and I know I can."
Feyenoord's hopes of challenging for the title were dealt a blow following the 2-1 defeat against leaders Ajax on Sunday, but a UEFA Champions League spot is not out of the question, with a four-point gap to second-placed Twente to make up.
And Rutten insists he has no intention on imposing on Koeman's job for the rest of the campaign.
He added: "The next few weeks are too important for Feyenoord to keep talking about me.
"I don't want to get in the way of Ronald Koeman and the squad or affect their concentration in any way.
"For that reason, I will stay in the shadows for the time being."
Technical director Martin van Geel is delighted to bring in an experienced replacement for Koeman, stating: "As people know, we preferred to have an experienced top Dutch coach for the first team after Koeman.
"Given all the demands we make on a coach the list of names you can draw up is certainly not too long. So it's great when you can reach agreement quickly with someone who is very high up the list.
"Fred Rutten is an experienced head coach who has demonstrated that he can improve players and often gets teams playing more attractive football.
"With that in mind, the club management agreed unanimously that he was the right man to take over from Koeman after a couple of good years."
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