The reform, hailed as the biggest shake-up of Mexico's telecoms and television market in decades, is aimed at spurring foreign competition and will give regulators the power to force companies to sell assets if they have more than 50 percent of the market.
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.
Shares of America Movil fell as much as 4.87 percent to 11.53 pesos, their lowest since April 2009.
"It's falling hard, aggressively and I don't doubt that there are people shorting the shares," said Gerardo Roman, head of trading at Actinver brokerage in Mexico City.
The stock has fallen sharply since Pena Nieto unveiled the bill on Monday. It tumbled nearly 8 percent on Wednesday alone, which wiped out almost $5 billion (3 billion pounds) of its market value.
The shares broke their losing streak on Thursday, aided by aggressive buying by the company's treasury, which purchased 150 million shares for a total of 1.837 billion Mexican pesos ( $147.71 million (97.74 million pounds)), according to two filings with Mexico's stock exchange.
Since the start of the week, America Movil's stock has lost some 13 percent, and is down more than a fifth in the year to date. It also suffered last month when the company released fourth-quarter results that fell far short of expectations.
Shares of Mexican TV giant Televisa 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 1655 GMT, Televisa shares were down 0.36 percent at 65.66 pesos.
Slim, the world's richest man, controls some 70 percent of Mexico's mobile market and 80 percent of its fixed phone lines. Televisa, controlled by Emilio Azcarraga, has about 60 percent of the broadcasting market.
($1 = 12.4363 Mexican pesos)
(Reporting by Dave Graham and Elinor Comlay; Additional reporting by Jean Luis Arce; Editing by Nick Zieminski and Bernadette Baum)