Rolls-Royce says it has "made progress" in its investigation into the cause of engine problems on the Airbus A380.
The firm did not say whether its engineers had identified what caused one of its engines to break apart on a Qantas flight last Thursday.
But the British firm said the problem was specific to the type of engine being used on the plane.
Shares in Rolls-Royce, which had fallen by about 10% since the incident over Singapore, rose on the announcement.
The firm's stock ended the day 2.7% higher.
On Monday, Qantas said it had found "slight anomalies" on three A380 engines and was keeping its fleet of six A380s grounded for further checks.
Tests have uncovered oil leaks in the engines of three of its grounded A380s.
Rolls-Royce said it was clear that the incident was specific to the Trent 900 type of engine which powered the A380 Qantas flight.
It said it had agreed a series of checks and inspections - which would allow airlines to resume flying the planes.
"We are working in close cooperation with Airbus, our customers and the authorities, and as always safety remains our highest priority," it said.
Other airlines using Roll-Royce powered A380s have continued to fly them without any major disruption.
Industry expert Sean Maffett suggested that this may be because the engines on the Qantas planes are slightly different.
He told BBC News that they have bespoke electronic controls which allow them to take off in hotter temperatures and at higher weights for longer distances.
This may explain why Qantas has still grounded its flights, he suggests. Rolls-Royce refused to comment.
All of Qantas's six A380 planes have Rolls-Royce engines.
The concern is that the engine failure could have been a sign of one or more major problems, which could hit Rolls-Royce's reputation and future sales.
Rolls-Royce, which is working with Qantas on the investigation, has seen its market value fall more than £1.5bn since the incident.
However, after falling by as much as 3.4% in early trading on Monday, Rolls' shares clawed back some losses after it announced a £350m service contract from EgyptAir.
The Airbus A380 superjumbo uses one of two engines.
The Rolls-Royce Trent 900 engine has been installed in more than half of the A380s currently in service.
The remainder use an engine manufactured by General Electric and Pratt & Whitney. Their engines have not been implicated in this recent safety scare.
The two businesses are committed to engines for a further 197 planes on order.
If it turns out that the Rolls-Royce engine is not fit for purpose and the engine is withdrawn, Rolls would have to offer a refund for the engines sold to its Airbus A380 customers.
What is more, Rolls Royce would lose the income that would come from servicing these engines - typically over 40 years - which makes up a big chunk of the engine-makers' revenue.
Shares in EADS, the aerospace giant that owns Airbus, have also been falling following last week's emergency landing by the Qantas A380. On Monday, its shares ended up 0.6%.