The musician, 74, is fully vaccinated and boosted, and is currently experiencing only mild symptoms, according to a post on his Instagram.
The global pandemic has caused many delays in Sir Elton's Farewell Yellow Brick Road world tour.
Another British superstar, Adele, also cancelled US shows last week.
The Dallas shows were originally scheduled to take place on Tuesday and Wednesday.
"It's always a massive disappointment to move shows and I'm so sorry to anyone who's been inconvenienced by this but I want to keep myself and my team safe," Sir Elton wrote on Instagram.
He assured fans they would be contacted once new dates had been announced.
He also said he expects to be able to perform at scheduled shows in Arkansas this weekend.