The reform, hailed as the biggest shake-up of Mexico's telecommunications market in decades, aims to allow more foreign competition and give regulators the power to force players that have more than 50 percent of the market to sell assets.
Late on Thursday, the lower house committee for constitutional matters passed the bill with no changes, and it is expected to be presented to the floor of the house for a vote next week. If approved, it will then head to the Senate.
Lawmakers in President Enrique Pena Nieto's Institutional Revolutionary Party (PRI) have said they are confident the bill will pass Congress before the end of April.
America Movil shares have fallen sharply since Pena Nieto unveiled the bill on Monday. The stock tumbled nearly 8 percent on Wednesday alone, losing almost $5 billion (3 billion pounds) (3 billion pounds) in market value.
The stock broke its losing streak on Thursday, aided by aggressive buying by the company's treasury, which purchased 75 million shares for a total of 916.6 million pesos (48.53 million pounds), according to a filing with Mexico's stock exchange.
Slim, the world's richest man, controls some 70 percent of Mexico's mobile market and 80 percent of its fixed phone lines, while Televisa has about 60 percent of the broadcasting market.
Televisa shares have also taken a hit from the planned reform, albeit to a lesser degree. They fell initially on Friday but later recovered losses to be broadly unchanged.
As of 1402 GMT, America Movil shares were down 2.3 percent, trading at 11.84 pesos. Televisa shares were trading at 65.9 pesos, down 0.05 percent.
(Reporting by Dave Graham; Editing by Nick Zieminski)