The Italy international – who was born and raised in Argentina – had been reluctant to move to St Mary’s amid interest from Tottenham and Inter.
The Corriere dello Sport and Sky Sport Italia report Osvaldo has been convinced to accept the move following a series of phone calls with Saints manager Mauricio Pochettino.
Roma had already agreed a transfer fee worth around €18m (£15.3m) with the Premier League side.
The Italy defender has been linked with several clubs over the summer, including Milan, Fiorentina and Southampton.
According to Sky Sport Italia, a deal has been struck between the two clubs for €13m-€15m (£11.1m-12.8m), depending on performance-related bonuses.
The player has been offered a four-year contract and the final details of personal terms are due to be discussed today.
Adem Ljajic has still not signed a Fiorentina renewal and it’s reported Atletico Madrid have beaten Milan’s offer for the sought after forward.
The Serbian striker’s current contract expires in June 2014 and the Viola have warned he will be sold this summer unless he pens a new deal.
Milan had been the favourites to snap him up, although their €8m (£6.8m) offer (plus performance-related bonuses) is still below Fiorentina’s asking price.
According to Spanish newspaper AS, Atletico Madrid have dived in with a proposal worth €10m (£8.5m).
Fiorentina released an angry statement criticising Milan’s approach towards Ljajic and would be very happy to sell him abroad rather than reinforce one of their Serie A rivals.
The Grifone need a new defender, having sold Granqvist to Russian club Krasnodar this week for around €5m (£4.2m).
According to the latest reports, Coach Fabio Liverani has targeted two Napoli players to fill the gap.
The prime target is veteran Gamberini, who is increasingly being pushed to the sidelines at the Stadio San Paolo.
The former Fiorentina man turns 32 in just over a week.
If reports about the imminent arrival at Napoli of Cagliari defender Davide Astori are true, then they may well be prepared to loan out Fernandez too.
Fernandez is a 24-year-old Argentina international who struggled to make an impact and was loaned out to Getafe for the last six months.
Inter are closer to taking midfielder Saphir Taider from Bologna after negotiations resumed in earnest.
The move had already been in the works several weeks ago, but seemed to stall with the Rossoblu unwilling to sell.
The Nerazzurri are still not matching the €8m price-tag, but the €6m (£5.1m) offer is close enough for the difference to be whittled down in talks.
According to the latest reports, Inter have proposed a four-year contract for Taider worth €800,000 (£680,000) per season.
If Bologna were to match that, it would put the 21-year-old among their top earners.
This week the Rossoblu brought back Diego Perez, who had left at the end of his contract last month, signing him to a new two-year deal.
Some have taken this as a sign Bologna are preparing to part with Taider.
Juventus forward Fabio Quagliarella is considering a new offer from Qatar club Al-Gharafa.
Sky Sport Italia claim the side, managed by Brazil legend Zico, are prepared to offer a €7m (£5.9m) transfer fee plus a three-year contract with impressive wages.
Quagliarella has been linked with several Premier League sides, most notably Sunderland, Southampton and Norwich City.
Al-Gharafa are building an impressive squad, as they have already signed Lisandro Lopez, PSG man Nene and former Parma and Palermo midfielder Mark Bresciano.