Lyon said the English club did not attempt to carry out the steps needed on world soccer governing body FIFA's transfer matching system for the deal to go through on Thursday before the end of the January transfer window.
"Following this incomprehensible behaviour... the international transfer certificate could not be requested before closing time," Lyon said in a statement.
"Olympique Lyonnais deeply regrets this situation and this decision, demonstrating a profound lack of respect on the part of West Ham towards the institution and the player," it said.
"The club reserves the right to initiate all appropriate procedures necessary to validate the transaction at a later date, and to hold West Ham responsible if necessary."
West Ham did not address Lyon's comments about why the transfer had failed but confirmed in an email to Reuters that the deal was not completed in time.
The club's manager David Moyes told reporters on Friday that the deal to take West Ham's Algerian forward Benrahma to Lyon had "some problems with the paperwork at the last moment".
"But I only learned about it when I came off the pitch late last night," he said.
Moyes said it was possible, however, that Spanish midfielder Fornals could still make the switch to Betis.
"I think there is a chance the Pablo deal (to Real Betis) will be ratified. I can't confirm if it will or not at the moment though," Moyes added.
Real Betis CEO Ramon Alarcon also said he hoped the Fornals transfer would get "resolved in the next few days".
"Betis sent all the documents correctly and on time and it seems West Ham had a computer problem," he told Spanish media.
"We imagine that with a computer error it can be resolved. We are working on it and we are optimistic," he added.
Sixth-placed West Ham visit Manchester United on Sunday.
