The Bayern Munich man missed yesterday's friendly win over Israel because of a calf problem which left him unable to train. However, Low has raised hopes the 27-year-old could be fit for Germany's opening match of the tournament against Portugal on 9 June.
"Schweinsteiger is now pain free again for the first time," he said in the Bild newspaper. "He can perhaps train again with the team on Monday or Tuesday."
Low admitted, though, his side have to improve when they start their campaign after the unimpressive showing against Israel in their final warm-up match. Germany, one of the tournament favourites, laboured to a 2-0 victory in Leipzig, their first win of the year following friendly losses to France and, last weekend, Switzerland.
Low said on dfb.de: "It was a reasonable final test. It will give us some impetus. Things are still not yet running completely smoothly, but to go into next week with a win does us good. In general we have to improve and take our chances. The organisation was better, but there are still things to do."
Goalkeeper Manuel Neuer added: "We produced a concentrated performance. It was difficult, because Israel were defence-minded. We found it tough-going, but we were clearly the better team."
Former Germany boss Berti Vogts warned Low that only winning the tournament would constitute success. Vogts, who won EURO 96 with the national team, said in Bild: "It was my job to win a title, I was an employee of the German Football Association. If is the duty of the Germany coach to win a title. It would be a shame, a real shame, if we did not manage that again."
That victory 16 years ago was Germany's last major championship win and Vogts said the wait needed to come to an end. He added: "The time has come to bring a title back to Germany again. At the moment we are the best team in the world. The players now have to prove it."