The survey was conducted between 24 July and 2 August and asked 2,576 likely voters in Texas who they would vote for if November’s election was held today.
The poll, from Morning Consult, showed that 47 per cent of likely voters in the state would vote for Mr Biden, while 46 per cent for the current US president.
Mr Trump won by nine points in Texas in the 2016 presidential election and a Democratic candidate has not beaten a Republican in the state since 1976, when former president Jimmy Carter was elected.
However, the presumptive Democratic candidate has targeted the state in recent weeks, as the president’s handling of the coronavirus pandemic has been criticised and Texas has fared badly with the outbreak.
The state has recorded its highest daily coronavirus total on multiple occasions over the last six weeks and postponed reopening parts of its economy in order to tackle the pandemic.
The poll also showed that Republican Senate incumbent John Cornyn, who has served since 2002, is leading Democratic challenger MJ Hager 44 to 38 per cent.
Last week, the Democratic Senatorial Campaign Committee (DSCC) announced that it would spend at least $1m (£761.415) in the state to boost Ms Hager’s campaign, and claimed the race was close.
DSCC executive director Scott Fairchild told Politico: “This race is a dead heat, and our increased investment reflects how MJ’s campaign and the increasingly competitive climate has put another offensive opportunity on the map.”
However, on Tuesday, in an interview with Fox Business host Lou Dobbs, the president claimed that recent polls that show Mr Biden leading in Texas and Democrats closing in on Republicans are not accurate.
“They’re against oil and gas. They’re against guns, and I guess they’re against God,” he said about the Democratic Party. “They’re against religion, they’re against the bible, certainly.”
He added: “How can you be against oil and gas and guns and lots of other things and you’re in play?” Mr Trump asked. “The polls are fake.”
Meanwhile, Mr Biden has seen his support in surveys rise in the same period and a Reuters/Ipsos opinion poll published late last month showed a lead over the president with registered voters nationally of eight percentage points.
Mr Biden had support from 46 per cent of those polled, in comparison to the president’s 38 per cent, while the remaining 16 per cent were undecided, according to the survey carried out between 15 and 21 July.
In an attempt to win in Texas, Mr Biden recently hired various staffers to help run his campaign in the state, including Jennifer Longoria, who served as senator Elizabeth Warren’s campaign director for her presidential bid.
After it was announced that the Biden campaign had made hires in the state, the Texas Democratic Party executive director Manny Garcia and deputy executive director Cliff Walker released a joint statement.
“By enlisting a state team of experienced Texans and making hard investment in the Texas Democratic Party’s coordinated campaign, the Biden campaign is treating Texas like the biggest battleground state that we are,” they said.
“Poll after poll shows Texas is up for grabs, with Joe Biden leading in most recent polls. Through unprecedented coordination, innovation, and smart investments, we will win.”