Messi's deal expires in nine days, and he has not yet made a public decision on whether he will extend his 20-year relationship with Barca.
President Joan Laporta remains optimistic Messi will sign new terms at Camp Nou, but Tebas said Barca must ensure they remain within the league-imposed salary cap.
"Barcelona have exceeded their wage cap," Tebas said on Monday. "I hope they can keep Messi, but to do so, they will have to make cuts elsewhere."
Prior to the COVID-19 pandemic, Barca had the largest cap in Spain's top flight in 2019-20 at €671 million.
However, La Liga's new limit for the 2020-21 season was €382.7m. The Catalan club were given some leeway in terms of reducing salaries, but it meant they were unable to register Memphis Depay despite agreeing on a fee with Lyon last September.
La Liga could technically prevent them from registering Messi's new contract or inscribing new signings this summer if they don't adhere to certain financial regulations.
Messi's most recent four-year deal, signed in 2017, was worth over €500m including his annual wage, signing on fees and bonuses.
Barcelona are still hopeful of selling players like Umtiti and Coutinho themselves. Martin Braithwaite, Junior, Neto, Miralem Pjanic and Matheus are other names who face an uncertain future at the club.
Tebas, though, said renewing Messi will be possible if Barca, whose gross debt stands at almost €1.2 billion, take measures this summer.
"The severity of the situation depends on the resources they are capable of generating," the La Liga chief added. "Barca have to restructure their debt. If they manage that, the situation won't be serious.
"But they have taken their wage bill to the limit. And when football has coughed -- or, better put, punctured a lung -- they haven't wanted to absorb that loss of income. They must take measures.
"Of the €700m losses [across La Liga], half of it is for Barcelona. There are other big clubs that have made a bigger effort to avoid it. Real Madrid have made a commendable effort to ensure their losses aren't the same as Barca, who take the trophy for losses."
On Sunday, Barca members approved a €525m loan from Goldman Sachs to help ease the club through a rocky period financially.
Laporta said the money would not be used on signings, but to pay outstanding debts such as player salaries, transfer fee installments and money owed to the tax office.
The loan can be repaid over 15 years and has a 3% interest rate.
ESPN revealed in May that Barca had taken an advance on the loan, prior to members approving it at the general assembly, to ensure they could make salary payments due before the end of the season.