Tracking Down the Rumor

I am not familiar with Trump's use of the term "I've heard", but I'm pretty sure he's not the only person who ever said that.  Although Trump is indeed a lying sack o' shit, that doesn't mean that everybody who uses some of the same words he uses is also a lying sack o' shit.  

Okay, the State of California is not allowing illegal immigrants to vote in state or federal elections, but they are allowing illegals to obtain driver's licenses.  When you get your driver's license in California, you are automatically registered to vote unless you opt out of it, which is probably how the rumor got started.  Also, you have to provide proof of citizenship before they register you, which should weed out the illegals.  Another way the rumor might have gotten started is:  "In November 2016, the California city of San Francisco did pass a ballot proposition allowing some non-citizens to register to vote in local school board elections only."  (My copy and paste did not work on this quote, so I wrote it down and typed it in myself.  It may be found at the bottom of the following article.)

https://www.snopes.com/fact-check/california-motor-voter-act/

But wait, there's more!
"A 1996 federal law prohibits noncitizens from voting in federal elections, but there is no prohibition on localities, and indeed a number of jurisdictions allow it, to some extent."
https://www.washingtontimes.com/news/2019/mar/8/house-votes-favor-illegal-immigrant-voting/