Carrick oversaw an impressive 3-1 win by United against Aston Villa on Sunday.
But when asked if Carrick should he given the job long-term, Keane said on Sky Sports: "I wouldn't. I think there's better options out there.
"I think the games where he's come in and in terms of winning football matches, he's done a very good job. He's obviously simplified things but there's been no jeopardy in those games.
"They've had three weeks to prepare for the matches. It's been good timing for him to go in there and he's done a very good job. I take my hat off to him and if he does get the job in the summer I'll say, 'Listen, good luck to you', but I think there are far better options out there for the next manager of Manchester United.
"If Carrick had been mentioned three months ago to be the next Manchester United manager you'd have been locked up. Everyone would've said he's not even in the reckoning but due to timing, he's got an opportunity and is doing a very good job."
United will qualify for Champions League
Keane continued: "I think there's better options out there, simple as that but he's done a very good job. Manchester United are certain to qualify for the Champions League now but in making decisions (on the next manager) they will have to go and interview people and see what's out there.
"Most players do like a caretaker manager. It's different when he comes in and has to make a decision about contracts, pre-season and medical staff, transfers and recruitment.
"His job in the last few months has been to win football matches. I know that's a manager's job anywhere but he's not had any distractions even as a manager. Of course he's done it and I give him credit for that but if he's given the job then there's different responsibilities for him.
"I think to manage Manchester United, you need someone a lot more experienced in terms of winning trophies, competing in Europe and he hasn't got that. He might have that in a few years, and he might be in the mix then, but the fact he's winning football matches at the moment of course gives him a great chance."
Lucky Carrick
Keane also stated, "All this chat that everyone loves him, I'm not so sure that was the case when they couldn't beat Newcastle with 10 men. He beat Aston Villa today - Aston Villa were absolutely shocking. The more I look at Aston Villa, the more I think they'll be missing out on Europe especially considering how bad they've been today.
"So the timing is good for United. For today's game Villa are on the back of a European game and United, because they were so bad at the start of the season, have no distractions. And remember this is the same group of players that finished 15th.
"So everyone else can get giddy about Man United being third in the league but I don't. I'm not one of them. I don't get all excited for stuff like that. I get excited when teams are winning trophies and competing week in, week out. So, I'm not one to get too giddy."