Ministers were this week accused of setting their sights too low, and introducing unambitious targets for education and training.
Baroness Kennedy told members of the Commons select committee on education that people should have the chance to study for A-level standard qualifications - not the GCSE-level awards currently proposed.
She was outlining the results of her report on widening access to FE at the first session of the committee's six-month inquiry.
Outlining the Government's commitment to the so-called "level 2" qualifications (equivalent to GCSE) for 16 to 19-year-olds, she said: "Certainly it's setting the sights rather low.
"To create an aspiration for all to level 3 (equivalent to A-level) is paramount. One of the things we are talking about is that education should be free for all people to the age of 19 or for adults without basic skills or who are economically deprived."
She was backed by David Melville, the Further Education Funding Council chief executive, who later told MPs: "We do make the assumption that the Kennedy students might go to level 2 - that's incredibly unambitious. Many of those Kennedy students will be going on to a degree." Baroness Kennedy told MPs that students from poor backgrounds, or those with no qualifications, should attract extra funding for colleges - perhaps twice as much as those from more privileged backgrounds.
But she went further, suggesting that the Higher Education Funding Council for England should be ordered to give priority to students from disadvantaged areas.
She insisted that further education should get extra funding - and hinted that extra cash may be taken from savings from the university sector, created by the advent of tuition fees.
Pressed by MPs on the possible redistribution of cash, she said Government should be "putting more money into FE than it has been getting in the past, and that probably means less money for HE.
"If HE is going to continue to grow and expand and to attract more working-class people ... it will have to look to FE as one of the areas which will assist in widening participation. It would be very easy for HE to see it being in its own interest to divert some of its money to FE because it will bring a greater number of students."
But Baroness Kennedy was anxious to point out that higher education had its own funding problems - and stressed that she was not advocating "robbing Peter to pay Paul."
Her report stopped short of recommending a transfer of cash into colleges from universities after fierce protests from vice chancellors, angry at the squeeze in funding within higher education.