The club have already agreed a deal to send Jesus Navas to Manchester City and therefore appear to be under no pressure to sell.
Sevilla announced on Monday night they had agreed a fee, reportedly worth up to £24million (28million euros), with City for winger Navas, who is yet to complete the move as he is on international duty with Spain ahead of the Confederations Cup.
At a press conference on Tuesday, Del Nido said the Navas sale was beneficial to all parties as Sevilla look to rebuild after a season he branded a "failure", but insisted he wanted the in-demand Negredo to remain at the club.
"He has four years left on his contract," Del Nido said of the 27-year-old striker, who scored 31 goals in all competitions this past season.
"He is at home here. He has scored a lot of goals and delighted us with his football.
"The summer will tell us in due course if he will still be with us, but if you ask my opinion I'd rather there were no bids. West Ham offered 17million (£14.5million) in cash with more in incentives but we do not value the offer and the player refused."
In contrast, it seems City more than met Del Nido's expectations in terms of an offer for their 27-year-old winger Navas.
"I think this is a great deal for both parties," he said. "They player had said personally to us and then publicly to the media that he wanted to leave. He believed his time in Sevilla was over having contributed all that he could.
"The price at which we have sold Jesus Navas is an amount much higher than we thought they were going to offer, especially as there was only one bidder."
With that transfer fee swelling the club's coffers, only similar high bids now seem likely to persuade Del Nido to part with others.
"We do not yet know the final figure (for Navas) as there are many variables and incentives in the deal," he said.
"But we have no need to sell as this has covered our budget deficit.
"In the past we have been willing to sell a player if the market value is offered, but so far all the offers that have come in have been rejected."
Sevilla's 4-3 win over Valencia on the final day of the season saw them finish ninth, taking the final Europa League spot subject to appeals from Malaga and Rayo Vallecano against bans which will prevent them from taking up European places.
However, Del Nido was still blunt in his assessment of their campaign.
"Let us make a distinction," he said. "In the (Copa del Rey) we had a remarkable run and reached the semi-finals before being felled by the ultimate winners.
"But in the regular season, the most important competition, we went backwards. You have to consider that we finished only ninth despite having the fifth or sixth biggest budget, and this indicates the goal has not been met."
Despite Del Nido's claims of a bid from the Hammers for Negredo, Press Association Sport believes the statement of a concrete offer is wide of the mark.
It is understood there was some tentative interest from the London club earlier in the season but no bid was lodged.