The 27-year-old Briton has starred in 2013, claiming a maiden pole at Assen and collecting four podiums from the opening nine races.
He is currently the top satellite rider in the championship, just one point behind Yamaha works rider Valentino Rossi and 47 in arrears to overall leader Marc Marquez.
Despite such success the Briton remains the only Yamaha rider without a deal for 2014, and has previously spoken of his desire to land a factory ride.
Lin Jarvis, managing director of the Yamaha Factory Racing, accepts he cannot offer Crutchlow a factory role, but still hopes he can persuade the Briton to stay with the Japanese marque.
"We would love to see Cal stay with us next year, but we realise that his performances are extremely good and that, therefore, he is very hot property and our competitors are also interested in his services," Jarvis told motogp.com.
"I am hoping he will stay with Tech 3 Yamaha. That will probably all be sorted before the next race - that is what I am expecting.
"We have a situation where Cal has evolved in Tech 3 and, normally, Tech 3 is our team where young or developing riders will come on and then maybe step up into the factory team.
"Cal is at the prime moment to step up, but we are full. We have Valentino [Rossi] and Jorge [Lorenzo] on two-year contracts so, right now, we have no place at the Yamaha Factory team.
"It is a difficult circumstance."
Ducati, with whom Crutchlow held talks for a 2013 ride, recently confirmed it would split with Nicky Hayden at the end of the current season.
Asked is he was aware of rivals' interest in Crutchlow, Jarvis said: "Yes, we are pretty aware of that. Two strong alternatives - and maybe both companies have more money than us at the moment, shall we say.
"Ducati especially have a different situation, so maybe they have more possibilities."
Asked how he hoped to secure Crutchlow for next year, Jarvis added: "Normally it means money. Basically, we are ready to do something.
"I think Cal is very happy with the team and with the bike and is very happy with where he is, but he is a professional rider and at a very important stage of his career.