The Saints pushed hard to sign the Dutch international. He has spent the last four seasons in Russia and helped Spartak win a league title last season.
A final offer of £25m was rejected by Spartak, who were holding out for £30m.
Pellegrino said on Promes: "It's a player that we had been following in the past but right now it's too difficult, financially.
"No, [they] didn't accept the bid. They didn't accept our bids."
The south coast club still hold hope they can complete a deal in the summer.