Updated at 10:30 a.m.: Revised to include comment from the Texas Democratic Party.

WASHINGTON -- Presidential hopeful Joe Biden has cemented his frontrunner status among the Democratic contenders in Texas, a delegate-rich early primary state that will see Houston play host on Thursday to the third round of Democratic debates.

Two polls released Wednesday show the former vice president leading the race in the Lone Star State to take on President Donald Trump in the 2020 election.

A Quinnipiac University poll puts Biden at 28% among Democratic voters and independent voters who lean Democratic in Texas, giving him a 10-point edge over Sen. Elizabeth Warren and a 16-point lead over Sen. Bernie Sanders and native Texan Beto O'Rourke.

A University of Texas/Texas Tribune survey, meanwhile, reflects much the same, with Biden at 26% among Texas Democratic voters, followed by Warren at 18% and O'Rourke at 14%.

"In Texas, the stereotype holds that Sen. Warren is the candidate of Democrats' hearts ... while Biden is the candidate of Democrats' heads," said Quinnipiac polling analyst Peter Brown, noting that respondents said Warren has the best ideas but that Biden has the best chance to beat Trump.

The survey results could spell trouble for the two Texans in the race: O'Rourke, a former El Paso congressman, and Julián Castro, a former San Antonio mayor and U.S. housing secretary.

While the duo have struggled to gain traction at the national level and in early primary states like Iowa, they've both pitched their ability to turn Texas blue for the first time in decades -- and win its 38 electoral votes -- in a general election showdown with Trump.

That Texas firewall is taking lots of heat.

O'Rourke remains near the top of the field in Texas, where last year he nearly upset GOP Sen. Ted Cruz in a high-profile race. He also fared better in two other recent Texas polls that showed him sitting in at least second place. But the pollsters noted that he's trending out of the lead.

Castro, in turn, appears to be an afterthought in his home state, polling at 3% in both of the new surveys.

The biggest change appears to be Warren's ascendance, which reflects her national rise. The Massachusetts senator picked up 7 points over a June Quinnipiac poll in Texas and 4 points over a June University of Texas/Texas Tribune survey in the Lone Star State.

She's seizing on that momentum, too.

Warren on Tuesday attracted several thousand supporters at a rally in Austin, where she railed against government corruption. Her appearance came a day after she waded into a Texas primary race by endorsing Jessica Cisneros, who's challenging longtime U.S. Rep. Henry Cuellar, D-Laredo.

But Biden remains the kingpin Democrat in Texas, as he does in the Democratic race as a whole.

"Senator Warren's popularity continues to grow in Texas as well as nationally," Brown, the Quinnipiac polling analyst, said in a news release. "But in either case, the changes are small, and Biden holds on to his double-digit lead."

No matter the intraparty dynamics, the Texas Democratic Party hailed the latest surveys.

Manny Garcia, the state party's executive director, seized upon the Quinnipiac poll, which showed that Trump's approval rating among all registered Texas voters is only 45% and that 48% of registered Texas voters said they would definitely not vote for the president in the general election.

"It has never been more clear: Texas is the biggest battleground state in the country," he said in a news release.

[BATTLEGROUND TEXAS? Click here to check out The Dallas Morning News' 2020 presidential candidate tracker, which shows where the contenders are visiting in the Lone Star State.]

The two new surveys also weighed in on other important issues in Texas.

The Quinnipiac poll surveyed Texas registered voters -- of all ideologies -- on gun control after deadly mass shootings last month in El Paso and Midland-Odessa. The survey found mixed feelings on the broader need for stricter gun laws and specific proposals like an assault weapons ban.

One exception: Nearly 90% of Texans expressed support for background checks for all gun buyers, the poll found.

The University of Texas/Texas Tribune survey, meanwhile, took the temperature of the crowded Democratic battle to take on longtime U.S. Sen. John Cornyn, a Republican who's been bracing for his toughest reelection campaign perhaps ever.

The results showed that the primary race is up for grabs.

Air Force veteran MJ Hegar led at 11% among Texas Democratic voters, followed by Dallas state Sen. Royce West at 5%, former Senate candidate Sema Hernandez at 3% and political organizer Cristina Tzintzún Ramirez at 3%.

Significantly, 53% of those asked about the race said they "haven't thought about it enough to have an opinion," while 13% said they simply didn't know who they would support.

The Quinnipiac poll, conducted Sept. 4-9, surveyed 1,410 registered voters, with nearly 460 them identifying as Democrats or Democrat leans. Its margin of error is plus or minus 3.1 percentage points for the broader survey and plus or minus 5.5 points for the Democrat-focused questions.

The survey used live interviewers calling landlines and cell phones; experts tend to consider that approach most effective and reliable.

The University of Texas/Texas Tribune poll, conducted Aug. 29 - Sept. 8, used an internet survey to query 550 voters who indicated their intention to vote in the 2020 Texas Democratic primary. It has an overall margin of error of plus or minus 4.2 points.