Findlay, Ohio-based Cooper has three plants in the U.S., including its largest in Tupelo, which employs some 1,200 workers.
The deal for Cooper is valued between $600 million and $800 million.
Apollo, according to the reports, is seeking to buy a majority stake in Cooper.
Forbes writer Taesik Yoon calls it a hostile bid.
"The language seems to imply that the acquisition attempt is not only unsolicited, it could be hostile," he wrote. "Usually when one firm acquires another it's for all of the shares outstanding. The fact that the source of the article specifically mentions what it would cost to acquire a 'controlling stake' is more consistent with a hostile bid than a friendly takeover offer.
Yoon said the deal would have to be closer to $800 million in order for Cooper shareholders to seriously consider a bid.
"This would place the total value of the firm at more than $1.6 billion, or roughly $25.50 per share," he wrote. "Let's round up to $26 for the sake of argument. This would represent almost a 40 percent premium over Wednesday's closing price."
Yoon goes on to say such a tender would be only 11 percent above the year-to-date high of $23.40 a share Cooper hit about a month ago.
Neither Cooper Tire or Apollo Tyres officials would comment about a potential deal.
"We have a strict policy of not commenting on rumors and speculation in the market," company spokeswoman Anne Roman told Reuters.
Shares of Cooper Tire jumped as much as 11 percent to $20.76 on Thursday, valuing the company at $1.3 billion. Apollo has a market value of about $840 million.
The acquisition of the world's ninth biggest tire maker by sales would give Apollo Tyres access to the U.S. market for replacement tires for cars and light and medium trucks.
According to Reuters, Apollo Tyres does not currently operate in the U.S. and gets two-thirds of its revenue from India. Truck tires account for about 60 percent of its sales in India and half of its global sales.
Cooper Tire focuses on replacement tires, an area of the market that is expected to benefit from growing fleets of aging vehicles in the United States, Samp&P Capital IQ analyst Efraim Levy said in the Reuters story.