The Proteas pace spearhead is nursing an ankle injury which saw him miss the recent series-deciding T20 against England and the subsequent two-match short-format games Ireland.
Rabada has now been wrapped in cotton wool ahead of the duel against Ben Stokes' side that kicks off at Lord's followed by matches at Old Trafford and The Oval.
Speaking to reporters ahead of a four-day warm-up game against the England Lions that begins in Canterbury on Tuesday, Elgar said Rabada has notably started bowling in the nets again but his workloads going into a Test match remain a concern.
"Obviously that is one of our top priorities that we have at the moment. Look, he has bowled since we've started our camp in Canterbury," the Proteas skipper said.
"Going through the workload I think that's maybe the biggest concern; whether he can carry himself throughout a Test match with the intensity and volume of overs in a day's play.
"For now he's doing all the right things, I can't say yes or no as yet because there are eight, nine days out before we play the first Test but it is high on our list to get him fit."
While Rabada is on the mend, South Africa have been significantly bolstered by the return to the Test squad of fellow paceman Anrich Nortje.
The 28-year-old played the last of his 12 Tests against West Indies more than a year ago and has been missing from the longest format due to injuries.
Elgar added: "It's great to have Anrich back. He's missed a bit with the injury over the last year. But the way he's bowling in the nets and white-ball games is big for us.
"He brings a whole new dimension to the table in terms of his pace. In the UK, if you have those assets, you need to use them."