Former vice president Biden has the support of 48 per cent of likely voters, compared with Mr Trump's 45 per cent, according to a Sunday poll by Dallas Morning News and the University of Texas at Tyler.
The same survey gave the incumbent a lead of 2 percentage points back in September.
If accurate, the poll indicates a small, but not insignificant, shift towards Mr Biden in a state that has not backed a Democrat candidate for the White House since 1976.
Buoyed by the news, Democrats plan to send Kamala Harris, Mr Biden’s running mate to Texas on Friday in an attempt to drive home the party's advantage in the Lone Star State.
Analysts say the shift in Texas may have been caused by Mr Biden's increasing support among the Hispanic community, whose 11.5 million residents make up almost a third of the state's population.
Due to his inability in recent weeks to make any inroads into Mr Biden's commanding national poll lead, the president's path to the White House has become increasingly more narrow than the one mapped out by his rival.
In 2016, Mr Trump secured almost 1 million more votes than then-Democrat challenger Hilary Clinton in Texas. Should he fail to hold onto the state, then it could all but end his chances of re-election and see Mr Biden claim victory.
Compounding the president's woes is that, according to the same Dallas News and University of Texas poll, some 40 per cent of respondents said they had already cast their ballots.
Due to the ongoing coronavirus pandemic, which has claimed some 18,000 lives in Texas, residents have been casting their votes early through drop boxes and postal services.
With just over a week to go until election day, this means the president has a smaller pool of voters to work with and attempt to convince before 3 November.
Traditionally, Republicans have hoovered-up votes in more rural areas of Texas, which has enabled the party to offset Democrats’ overwhelming support in the state’s urban centres such as Dallas, Austin, Houston and San Antonio.
But tens of thousands of Americans have, in recent years, been flocking to growing cities such as Frisco, New Braunfels and McKinney in search of work.
These voters, who tend to be more liberal, added to the state’s growing Hispanic population — who also tend to favour Democrats — could see Texas choose a non-Republican presidential candidate for the first time since Jimmy Carter.