The Alexandria City Council last night approved the $16 million sale of the city's financially troubled cable television franchise to a subsidiary of the billion-dollar, Chicago-based Tribune Co. Inc.

The present operator, Alexandria Cablevision Inc., a locally owned company, announced this fall that the cost of paying high interest rates would force it to sell the franchise, which extends until 1994.

The Tribune Co. was selected by Alexandria Cablevision from among at least three bidders, all of them major cable operators. Officials of the Alexandria firm said they chose the Tribune Co. even though its $16 million bid was not the highest received. Alexandria Cablevision shareholders stand to make a fivefold return on their original investment..

Alexandria cable regulations required the city to approve the company's selection.

Under provisions of the franchise contract agreed to last night by the council and officials of Tribune Co., the 10,000-subscriber system will be completed by Dec. 31, 1982, a year ahead of present schedule. The new franchise, called Tribune Company Cable of Alexandria, also agreed to a one-year freeze of the current $9-a-month basic subscriber rate.

Officers of the new company also agreed to purchase $85,000 worth of production equipment to be used by the city, and to maintain the city's local origination channel, Colonial 10. Tribune Company Cable also promised to provide by the end of next year one additional pay service offering special entertainment programs for an added fee, and to add one new basic service, such as a cultural channel, by the end of next year.

A controversial feature of the contract is a provision, stating who must pay for installing cable underground, that was rewritten last night at the suggestion of Republican Vice Mayor Robert Calhoun, who called the original wording ambiguous.

As redrafted, the provision apparently will benefit only two citizens -- Republican State Sen. Wiley Mitchell and his neighbor in the Clover section of the city, former Alexandria Hospital administrator Charles Goff, now a real estate agent.

Mitchell and Goff had wanted the cable company to be required to pay the estimated $300 cost of running underground cable to their homes. The council approved a compromise providing for the company to pay up to $150, or half, of the cost of running cable to homes where the homeowner was not required by the city to bury utility wires.

Councilman Donald Casey, a Democrat, cast the lone dissenting vote on the provision, which he referred to as the "Wiley Mitchell protection act."