More specifically, we could use our own engine. Gripen E still rely on the GE F414. Europe has nothing to rival the P&W F135.
My limited understanding is that the F135 is massively over-powered to be capable of VTOL + push through the bulky shape of the F35, resulting in a disappointing range. I don't know that it would make sense to use it on a different platform.
I was thinking more in terms of it being a newer design.
From what I've gathered there's some work being done on new engine designs within the FCAS program, but I have no idea how it's going.
That just means it's the right size engine for a bigger plane without vtol.
The Gripen could have been designed around one Eurojet EJ200 or Snecma M88.
No longer: Gripens are now offered with the Rolls Royce EJ230 engine.