Liverpool have announced that they have agreed terms with Barcelona over the sale of Javier Mascherano.
Mascherano, 26, has been desperate to leave Anfield due to family reasons, and he has refused to play in Liverpool's last two games as he looks to force through a move.
Inter have also made an approach for the Argentina captain, but Hodgson said on Thursday night that he considered the offer to be derisory.
"I saw a fax the other day but I think it was sent tongue-in-cheek, to be honest," he told Sky Sports. "The offer was so unbelievably far from a valuation it seemed to me the sporting director was putting the fax in to maybe satisfy people at the club. "We didn't even bother to answer it - that is how ridiculously far from a serious offer it was."
It is claimed that Inter offered just £6 million plus former Portsmouth midfielder Sulley Muntari.
Liverpool have now confirmed that they have agreed a fee with Barca, reported to be an initial €16 million rising to a potential €22 million.
A statement on the club's official website read: "Liverpool Football Club have agreed terms for the transfer of Javier Mascherano to FC Barcelona. The Spanish club have now been given permission to speak to the Argentina midfielder."
A Barca statement added: "FC Barcelona and Liverpool FC have reached an agreement in principle for the transfer of Javier Mascherano to the club for the next four seasons.
"The deal was closed after intensive efforts made by the club in the final hours and through the player himself in the negotiations. Barcelona will report the details of the operation once the player passes the medical and the transfer is concluded. It is hoped Javier Mascherano will arrive in Barcelona this weekend."
Liverpool club secretary Ian Sylvester said Hodgson would be looking to reinvest the money before the end of the month.
"Mascherano is one of the best midfielders in the world and he has decided to go, and I'm sure Roy would not want one of his best players to leave without anyone coming in," he said.
"There's still a few days until the transfer window closes. We need to improve on last season domestically, there's no doubt about that. We want to get back into the Champions League."