In a regulatory filing, American Airlines said Qatar intended to buy at least $808m (£638m) of its shares.
American Airlines said it would respond "in due course".
Shares in American Airlines, which is the biggest airline in the world, initially rose 5% but fell back to a 1.2% rise by afternoon trading.
Qatar Airways said it sees American as a good investment opportunity and would not involve itself in management, operations or governance.
"Qatar Airways believes in American Airlines' fundamentals," the firm said.
In 2015, Qatar Airways took a 10% stake in International Airlines Group (IAG), the owner of British Airways and Iberia.
Then in July 2015 it increased its stake to 20%, becoming IAG's biggest shareholder.
But the airline has encountered turbulence.
Earlier this month, Saudi Arabia, Egypt, Bahrain, the United Arab Emirates, Libya, Yemen and the Maldives cut diplomatic ties with Qatar.
Saudi Arabia, the UAE, Bahrain and Egypt all said they would stop flights in and out of Qatar, and close their airspace to the country's flag carrier, Qatar Airways.
US companies, including American Airlines, have also pressured the White House to take action against Qatar Airways and other Gulf airlines, which they say receive unfair government subsidies.
Seth Kaplan, managing partner at Airline Weekly, said Qatar Airways may be hoping an investment would help reduce those objections.
The firm may also be looking to strengthen Qatar's relationship with the US, which has appeared uncertain amid the diplomatic dispute in the Gulf, he said.
"It's almost certainly not just a financial investment," he said.
American Airlines and Qatar Airways are already partners through the Oneworld alliance. This investment could tighten that relationship, but it is unlikely to have a large effect on consumers, Mr Kaplan said.
It is not clear at this point how American Airlines will respond, Mr Kaplan said.
In its statement, Qatar Airways said it would not acquire more than 4.75% of shares without permission from the American Airlines board.
American Airlines has received an "unsolicited" approach from Qatar Airways which wants to acquire 10% of the US carrier.