Myler signed a one-year deal in July to join the Swansea-based side with Wales and Ospreys playmaker Gareth Anscombe out with a long-term knee injury.
The 36-year-old has been in impressive form since joining from London Irish and has had a major influence on the young Ospreys players around him.
He has also been reliable with the boot in the Guinness PRO14, kicking 18 of his 19 penalty attempts for a success rate of 95 per cent.
On whether Myler might stay another season, Booth said: “Our intent is certainly to try and keep him here, I think he has done enough to have those conversations.
“The team predominantly is the same and he has literally been the quarter-back of the improvements at the Ospreys.
“So that warrants a conversation which is going on now and if we could firm that up it would be awesome. It’s got to be right for Steve too.
“I’m sure people raised a few eyebrows when Stephen walked through the door at the Ospreys at 35 but look what he’s doing for us. He defies age.”
