Yes, but why!? Rolls-Royce were the 'establishment' who would have an interest in maintaining the current piston engines, or at least Whittle could perceive that. It was the 'establishment' he been fighting against all the years prior. However, after working with Rover reality kicked in.
To quote from W. Freeman's biography (p262 - 264):
Freeman spent the first part of the meeting (11 Dec '42) trying to persuade Whittle to accept the logic of throwing in his lot with Rolls-Royce as their jet-engine designer, but although Whittle was delighted at the thought of Rolls-Royce taking over the Rover factories and advancing the deveolpment of his engine designs, he obviously hated the very idea of R-R taking over his company.
Apart from his prejudice against the establishe aero-engine industry, there were three other likely reasons for Whittle's resistance to a formal association between the two companies.
Firstly, Power Jets was both his 'command' in the RAF and the 'fons et origo' of all British gas-turbine inventions: jet-powered flight had been achieved with his engine, and his later designs were full of promise. He wanted to be given space machinery and labour to originate new engine designs, to make the first batches of each new engine, and to continue to be the designer, developer,and most of all, the Chief-engineer of the enterprise. Secondly, acquisition by Rolls-royce would leave no role for three other power Jets directors, Johnson, Williams and tinling, who had given valuable non-technical help to Whittle since 1936, and to whom he was staunchly loyal.
Thirdly, there is little doubt that Whittle's own mind was more than a little unbalanced by this time, for he had become addicted to the frequent use of 'benzedrine' inhalers, which could be then purchased from any chemist. The amphetamine suphate constituent of the drug was not then recognized to be addictive, but the side effects are now known to cause insomnia and feelings of anxiety and, at times, parania.
Footnote to this last paragraph:
Bulman's memoirs explained that Whittle became increasingly suspicious and unreasonable as a colleage during the period 1940-42.
Therefore for an alternative scenairo to have occured to by-pass (pun intended) Rover, it would have needed a combination of a number of factors - some personal contact with R-R people to show Whittle that his perceptions are not well founded - perhaps contact with other great designers such as Barnes Wallis to show it was possible to work in a big company, more support from the Air Ministry in general rather than mainly Freeman, and a bit of far sightedness from Beaverbrook (though I know that's asking a lot).