Free agent Juric was on the Saints' list to replace Ralph Hasenhuttl two years ago.
He was second choice to eventual successor Nathan Jones.
Southampton have also sounded out West Brom boss Carlos Corberan and former Denmark coach Kasper Hjulmand about succeeding the recently axed Russell Martin.
Sheffield Wednesday boss Danny Rohl was considered a frontrunner for the job but his £5million compensation fee became a stumbling block.
Corberan was also keen on the job but Saints were reluctant to meet his buyout clause - which is understood to be at least £2.5m.
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Croatian manager Juric, who is in England today, has been out of work for just over a month.
The 49-year-old has a style of play that has been described as 'heavy metal football', involving working his players hard in training and placing a big focus on running.
He was sacked by Roma after just three months in charge after taking over from Daniele De Rossi.
Juric left Roma 20th in the Europa League table and only four points ahead of the relegation zone in Serie A.
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He will have another relegation battle on his hands at St Mary's, with Southampton currently bottom of the table and nine points from safety.
Former Sevilla and Hajduk Split midfielder Juric first went into management in 2014 at Italian outfit Mantova.
Spells with Crotone, Genoa, Hellas Verona and Torino followed.
He parted ways with Torino at the end of the 2023/24 season after three mid-table seasons with the Serie A side.
Roma appointed him as their manager in September after De Rossi picked up just three points in their first four games of the campaign.
However he failed to turn their fortunes around and was dismissed after 12 games in charge.
He managed four wins, three draws and five losses in that time.
Southampton sacked Martin after being thumped 5-0 by Tottenham at home last Sunday.
The 38-year-old had guided the Saints to promotion via the play-offs from the Championship last season at the first time of asking following their relegation the year before.
But any credit in the bank ran out when his side suffered their 13th league defeat in 16 games against Spurs.
Southampton have scored just 11 goals this season in the league, conceding a whopping 36. Only Wolves have conceded more, letting in 40 so far.
The Saints suffered a 2-1 defeat to Liverpool in the Carabao Cup quarter finals on Wednesday in their first game without Martin.
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They travel to Fulham on Sunday in the Premier League as they look to claim their second league win of the season.
Their only other victory in the league this campaign came in a 1-0 win over Everton at home at the start of November.
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