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Town have stiff competition though as fellow recently-relegated League Two side Chesterfield are interested, as well as now-League One outfit Rotherham United, that's according to Andy Giddings, the sports editor of BBC Sheffield.
O'Grady is currently a free agent following his release from Brighton, where his career stumbled, scoring just one goal in 31 outings for the Seagulls.
Since joining Brighton in 2014, O'Grady has had three loan spells away from the AMEX - at Sheffield United (one in four), Nottingham Forest (two in 21) and most recently, Burton Albion last season (one in 26).
One of his best seasons came at Barnsley, in the Championship, in the 2013-14 season when he netted 15 goals for Tykes in 40 matches, over double his tally of the previous campaign (six in 15).
O'Grady has also spent time at the likes of Rochdale, where he scored 31 goals in 90 games over a three-season period.
New manager David Flitcroft, who brought O'Grady to Barnsley, will be in sole charge of recruitment at Swindon Town and chairman Lee Power has said the Robins will have one of the healthiest budgets in League Two this season.
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