Coach Jones is under pressure after he watched his England side conclude a dismal autumn series with Saturday's 27-13 loss to South Africa, who were without their European-based players but still inflicted a sixth defeat of 2022 on the hosts at Twickenham.
It completed England's worst year since 2008 and the Rugby Football Union is now conducting a review, which will inevitably decide if Australian Jones is to continue until next year's World Cup as planned or will leave his post 12 months earlier.
RFU chief Bill Sweeney admitted "results are not where we expect them to be" and Woodward, who masterminded England's solitary World Cup win 19 years ago, was yet again fierce in his criticism of his old rival Jones after the match.
Woodward labelled rugby in England as a shambles and insisted the weekend defeat was "one of the most depressing games I've seen at HQ".
Jones, who has already agreed to step down after the 2023 World Cup in France, told Men's Health UK: "I feel sad for him (Woodward). If that is the best thing he has to do in his life, then he hasn't a lot to do."
The pair's rivalry stretches back a long way, with Woodward having coached England to their World Cup final win in 2003 over Australia, then led by Jones, after a tournament in which they sniped at each other.
"I'm 62 now and in pure coaching terms I am coaching better than I ever have. Results aren't always perfect, but I'm happy with how I have been coaching," Jones said.
"After this, I want to do something really meaningful. I've enjoyed England a lot, it was a bit of a rescue job at the start, now rebuilding, and I am confident I will leave things in good shape."
England were booed off at Twickenham after defeat to the Springboks, which meant their record for 2022 finished at five wins, one draw and six losses.
Scotland, Ireland, France, Australia, Argentina and South Africa have all beaten the World Cup runners-up during the past 12 months but Jones believes they can still be a force at next year's tournament.
Currently the fifth favourites to lift the Webb Ellis Cup, England will avoid the front-runners until the semi-final stage, with one of Australia, Wales or Fiji a probable last-eight opponent in France.
"If this was the Cheltenham Gold Cup, there's a pack of four out front - France, Ireland, South Africa, New Zealand - and we are fifth, right behind them, right on the rails," Jones said.
"A good position, provided we keep improving.
"Australia are there or thereabouts with us. It's going to be the closest World Cup ever. France and Ireland are the in-form teams right now, but things will change."
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