The Netherlands defender expects to be a Reds player by the end of the transfer window.
Liverpool have been linked with a move for Van Dijk all summer.
Hoedt joins the Premier League side for a reported fee of £15m.
Howedes has emerged as an alternative to Virgil van Dijk for Liverpool.
Costa has set his sights on a return to Atletico Madrid while Van Dijk's future is in limbo.
Lazio sporting director Igli Tare confirms that Hoedt will join the Saints "in the near future".
Gazzaniga spent the 2016/17 season on loan at Rayo Vallecano.
Southampton have reportedly offered £15m for Lazio's Hoedt.
Liverpool and Chelsea have both been linked with Van Dijk throughout the summer.
Ex-Reds boss unsure whether Virgil van Dijk can cure the club's defensive issues.
Blues hopeful of having a bid of around £20m accepted for Cedric.
Gao and his daughter, Nelly Gao, buy controlling stake in Premier League club for £210m.
Transfer window could be cut short by around three weeks next season.
Oxlade-Chamberlain has one year left on his contract at Arsenal and has not signed a new deal.
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Van Dijk has released a statement stating he wants to leave.
Van Dijk submitted an official transfer request earlier this week.
Another eventful summer on the south coast sees the former Liverpool coach take the reins from Claude Puel.
Van Dijk submitted a transfer request on Monday (7 August) after growing frustrated by Saints' stance.
Wright wonders if Arsene Wenger will pull a rabbit out of the hat before the end of the transfer window.
Dutch defender has released a detailed statement clarifying his position after troubled summer.
Gabon international poised to move to the St Mary's in £20m deal.