Smith told reporters on Sunday that he would be "struggling" to be fit in time for the start of the four-Test series which begins at Lord's on July 10 after tearing his hamstring while playing in Indian Premier League.
Arthur however said that although the injury usually takes four weeks to recover he was hoping Smith would be available for the final warm-up game.
"It's a grade two injury, which normally means about four weeks out of action but I am confident (physiotherapist) Shane Jabaar will get him ready in time," Arthur told the Sapa news agency.
"Graeme will probably miss the first warm-up match against Somerset and we will be looking at the warm-up against Middlesex, starting on July 4, as Graeme's comeback game. I'm not even considering Graeme not playing at Lord's."
Smith has fond memories of playing at Lord's, scoring a match-winning 259 on his last appearance in north London in 2003, shortly after becoming captain.
Arthur meanwhile said he was confident that star batsman Jacques Kallis, who has also been playing in India, would be fit for the series.
"Jacques Kallis picked up a bit of a niggle towards the back end of the IPL, but he's also doing fine. I firmly believe that by the time we leave for the United Kingdom, we will have a fully fit side," he said.
The South Africans are due to hold a training camp in Johannesburg before flying to England for a tour which will also see them play five one-dayers and a Twenty20 international.