I sit on the standards body for Java EE which will be involved in helping make this step - would love to hear opinions on what foundation and licensing they'd like to see this under (and why).
Some Java EE specs are already at Eclipse (long standing historical reasons), so there is a small precedent
Anything open source, non-GPL is fine. It's important to have the TCKs open source too. We can't continue with this circus that Tomcat and Payara do not get a TCK and can't call themselves JavaEE compliant.
I'd have a small preference for Eclipse, mainly because some specs are already there (like EclipseLink).
Apache already has their own implementation of much of Java EE (via the Geronimo project, TomEE and Tomcat, among others).
In my opinion it would be slightly confusing to have say 2 JSF implementations at Apache (Myfaces and Mojarra). Not really terrible, but my gut feeling says Eclipse would be a better home.
It's important to note that Oracle can't do this on their own. The IP of the specs is with the spec lead so every spec lead has to agree to a transfer of their IP.
Some Java EE specs are already at Eclipse (long standing historical reasons), so there is a small precedent