Horne had been replaced by understudy Fergus March after he collapsed at a matinee performance last Thursday. Staunton was also advised to rest and was replaced by Sharon Eckman, What'sOnStage reports.
In a statement, Horne said: "I am delighted to say I am feeling much better and will return to the show tonight for the final week.
"I feel terrible for letting audience members down last week when I was ill, and I am so grateful for the wonderful letters of well wishes from everyone."
He added: "We have all had an amazing time working on Sloane since January, and I am just glad to be back in full health for the final week of this tremendous production."
Horne's agent Rupert Fowler said that the actor was suffering from exhaustion because of his busy work schedule, while a theatre insider claimed that he had "flipped" over recent criticism.
Entertaining Mr Sloane closes on April 11.