The competing trips to southern Texas come as border security is now a top issue for voters, with 17% of Americans listing illegal immigration as the most important problem facing the country, according to a recent Reuters-Ipsos poll.

"This is a Joe Biden invasion."

In Eagle Pass, Texas, Trump sought to capitalize on the rising concern among Americans about immigration and to pin Biden's name to the issue.

"It's migrant crime. We call it Biden migrant crime. But that's a little bit long, so we'll just leave it. But every time you hear the term migrant crime, you know where that comes from."

In Brownsville, Texas, Biden sought to shame Republican lawmakers for rejecting a bipartisan bill to toughen immigration policies after Trump told them not to pass it to avoid giving Biden a policy victory.

"The majority of Democrats and Republicans in both houses support this legislation until someone came along and said, don't do that, don't benefit the incumbent. That's a hell of a way to do business in America for such a serious problem. We need to act."

Under pressure from Republicans who accuse him of failing to control the border, Biden called on Congress last year to provide more enforcement funding and said he would "shut down the border" if given new authority to turn back migrants.

The number of migrants caught crossing the U.S.-Mexico border illegally hit a monthly record of 250,000 in December but dropped by half in January, a trend U.S. officials attribute to increased Mexican enforcement and seasonal trends.