A number of supporters said on Twitter they had been targeted by racist abuse from other fans at the Birmingham venue on Monday.
Former Yorkshire spinner Azeem Rafiq shared a thread detailing some of the incidents and wrote that they were, "disappointing to read".
Rafiq's Testimony to a British parliamentary committee last year led to an investigation into claims of institutional racism at Yorkshire and resulted in major reforms.
Edgbaston's official Twitter account responded to Rafiq's tweet saying that they would investigate the matter as soon as possible.
The England & Wales Cricket Board said in a statement it was "very concerned" by the reports.
"I'm gutted by these reports as we're working hard to make Edgbaston a safe, welcoming environment for all," Stuart Cain, chief executive at Edgbaston, said on Monday.
"Having seen the initial tweets, I've spoken personally to the gentleman who raised them and we're now speaking to the stewards in this area to establish what happened.
"Nobody should be subject to any form of abuse at Edgbaston. So, once we've got all the facts, we will make sure this issue is addressed swiftly."
The Bharat Army, India's official global supporters' group, later said many of its members had been targeted by "a very small minority".
"We will work with @Edgbaston to share all your feedback. Thank you to those England fans who stood by us," it added.
The rearranged fifth Test will resume on Tuesday, with England needing 119 runs on the final day for a series-levelling victory.