Wot they said, plus....
It sounds like you have a leak in the tyre bead seat - between the tyre and the wheel, where they meet. A good tyre fitter can seal that joint to stop a leak - the tyre has to come off to do it, but we're talking a total of under £20 for one wheel.
If the problem is that your alloy wheels are now getting old and starting to corrode, you might start having this problem on all your wheels, even when new tyres are fitted. So make sure the tyre fitter knows about this and has a plan to deal with it. It does not affect the runflat/non-runflat issue in any way.
The long term answer, like in the next year or two, if it is corroding alloys that is the problem, is to get the wheels refinished. That's likely to be in the region of £50 per wheel, but of course you do then also get wheels that look 'as new'.
You need to ask around your mates for a decent tyre shop. It will probably be an independent or small local chain, as most of the big boys are ***** at best and crooks at worst. There will be somewhere good and it's worth finding them.