Northampton pair Ben Cohen and Steve Thompson have already opted out of the two-Test trip, while their fellow World Cup-winner Josh Lewsey is also unlikely to travel and Robinson has previously pointed towards captain Martin Corry enjoying a summer breather.
Elsewhere, props Andrew Sheridan and Matt Stevens are both long-term injury victims, and there is a doubt surrounding Bath lock Steve Borthwick, a leading candidate to skipper the squad if Corry stays at home.
Borthwick watched Bath beat Worcester in their Premiership finale at the Recreation Ground after undergoing minor knee surgery following his club's European Cup semi-final loss to Biarritz last month.
"I couldn't keep playing on it, but now any involvement in the tour is fully dependent on the knee being in top condition," he said.
"The prediction given by the surgeon was that it would be borderline. I'm doing everything I can to get the rehab right.
"I want to go on the England tour, but the worst possible thing would be for problems to recur and also compromise my pre-season. It is up to me - I am the one who knows my body best - but no-one from England has spoken to me yet."
Sale Sharks coach Kingsley Jones has revealed Sheridan could face an operation which would rule him out until the start of next season.
Sheridan will receive the results of a scan on his damaged neck disc early next week, and Jones admits it could result in the prop forward undergoing surgery.
"We will get the results in a couple of days, and if he needs an operation that could mean him missing the first one or two matches of next season," said Jones.
Sale intend leaving a decision on whether fly-half Charlie Hodgson and wing Mark Cueto journey Down Under for Tests against Australia on June 11 (Sydney) and 17 (Melbourne) up to the players and Robinson.
"They have been playing non-stop rugby and I don't know whether they need to go, but it is up to them to discuss it with Andy Robinson," Jones added.
Northampton coach Paul Grayson backed Cohen and Thompson, who boast exactly 100 England caps between them, following their decision to miss the tour.
"They made the decisions for themselves," said Grayson.
"Having the summer off will deliver them fully fit for us next season and enable us to deliver them fully fit for England going into World Cup year."