The Colombia forward scored two stunning goals to help Atletico beat Athletic Bilbao 3-0 in the Europa League final on Wednesday, finishing the tournament as top scorer for the second season in a row, with 12 goals.
Atletico lie fifth in La Liga, two points short of fourth-placed Malaga in the last Champions League qualification berth, and visit relegation-threatened Villarreal in the final round of matches on Sunday.
"If the club don't manage to qualify for the Champions League they are going to have a problem in terms of income," Gil Marin told Spanish radio. "Atletico need to be in the top four.
"If we don't make it, it will be difficult that he (Falcao) stays."
Falcao was signed from Porto for £35 million last year after he helped them win the Europa League, and has quickly found his feet at the Calderon.
Atletico used some of the money raised from selling Argentine Sergio Aguero and Uruguayan Diego Forlan, who helped Atletico win the inaugural Europa League in 2010, and the 26-year-old has been their leading marksman with 23 goals in La Liga this season.
Only Barcelona's Lionel Messi and Real Madrid's Cristiano Ronaldo have scored more, but Atletico's earnings are only around a quarter of Spain's big two, and their lack of financial clout may force them to cash in on their prize asset.
"(Fernando) Torres went and we carried on winning, and Aguero went and we continued winning," Gil Marin added.
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