Arduino 12v LED Downlights
The output of the "transformer" you refer to is 12VDC - ie it has rectification included so the LEDs are driven by DC. 



So "dimmable LEDs" probably just means "accepts variable voltage DC". If so you should be able to use your existing circuit (which you should add to your post or provide a link to) with the "transformer" DC output, where V+ is now the +ve feed from the 'tranformer' and the LED downlights connect where the LED strip is now. The circuit  that Emmanuel posted in his now deleted answer should also work and is likely to be functionally the same as your one. .

With the above arrangement, when you provide full on drive from the Arduino the LEDs will be dimmable manually, and when you provide full brightness with the manual dimmer the LEDs will be dimmable using the Arduino. Dim them both at once and they will interact - but it should do no harm.