Boss Sean Dyche says Rodriguez, who missed Monday's 1-0 win at Crystal Palace due to a hamstring issue, has a chance of being available, along with Johann Berg Gudmundsson (calf), while Robbie Brady is a "maybe".
All three midfielders missed Thursday's impressive win over Tottenham, Lundstram with an arm injury and Fleck and Freeman with unspecified problems, and boss Chris Wilder has ruled them out for a couple of weeks.
Jack O'Connell is still battling a knee injury, but is back training on the grass.
Provisional Burnley squad: Pope, Peacock-Farrell, Jensen, Taylor, Tarkowski, Mee, Pieters, Bardsley, Long, Dunne, Thomas, Gudmundsson, Brownhill, McNeil, Brady, Westwood, Goodridge, Rodriguez, Vydra, Thompson.
Provisional Sheffield United squad: Henderson, Moore, Verrips, O'Connell, Robinson, Basham, Egan, Baldock, Stevens, Berge, Norwood, McBurnie, Sharp, Mousset, Zivkovic, McGoldrick, Jagielka, Rodwell, Osborn.