Speaking on the Hay Zoom programme, Tapia said the agreement was reached several weeks ago and had been kept under wraps deliberately, with the aim of waiting for clarity around the ANFP presidential elections.
Asked about the claim, Real Betis said they had no knowledge of any such agreement and pointed instead to the manager's contract, which runs for a further year after his renewal was confirmed last November.
Pellegrini, who arrived at Real Betis in the summer of 2020, is contracted until at least June 2027, with any further extension still to be determined.
According to Tapia, there has been no formal declaration on the matter.
"Formally there's no statement; this is a verbal agreement reached with Manuel Pellegrini through work done by Felipe Correa, the federation's national teams manager," said Tapia.
"Everything to do with the national team was kept separate from the ANFP elections, and Pellegrini asked for clarity on the administrative side of how the separation between the ANFP and the federation would be handled."
The ANFP, Chile's football federation, is in the midst of a power struggle that will culminate in elections in three months' time, following the end of current president Pablo Milad's term.
Milad took charge in 2020, and Pellegrini's name has been mentioned on several occasions as a potential candidate to take over the national team, which has been in a deep crisis for years and failed to qualify for the World Cup currently taking place.
The Chilean coach has guided Real Betis to European qualification in every season of his tenure, this year securing a Champions League spot after finishing fifth and taking advantage of an extra berth awarded to Spain through the coefficient system.
He also won the Copa del Rey with the Andalusian club in 2022.