Short answer: No, Parliament never appeared to seriously extend any sort of representation to the Colonists during the prelude to the revolution.

Long Answer This question is similar to one that was just asked a couple of weeks ago, where someone asked if the Colonists were justified in their rights to rebellion, so I will post my answer to that here

That said, Parliament during the era never seriously seemed to consider extending any type of representation to the Colonies in the 1760s or 1770s. They believed (and rightfully so) that the Colonies were too dependent on Great Britain and that they would not actually risk open rebellion against them. And even if they did risk military action, they'd simply crush them. Anything else just didn't seem logical. In many ways, they were right. The Colonial gentry were not all on board to want to separate from Great Britain, even by 1774 (the year before the battles of Lexington and Concord) -- especially in the south. Southern plantation owners were HIGHLY pressing their representatives to avoid all conflict with Great Britain... Well, until the local people started to rebel against them.

Recently, many historians have started to show how it was the lower-classes that began fighting back against their injustices years before the Revolution even started. A great book on this is Breaking Loose Together: The Regulator Rebellion in Pre-Revolutionary North Carolina by Marjoleine Kars. The book tells the story of rural farmers who were tired of the local gentry that stood by as the people down below suffered. What's funny, is that this is one of many tales where this happened in the pre-war era, and it is believed by some historians (myself included) that it was the rise of the peasants and their anger that pushed the gentry to go to war (they saw these rebellions as terrifying, and they realized they needed to catch up or they'd be lost).

The actual phrasing "No taxation without representation" was a mantra of the working-class. These were the people who were doing the "rough music" -- the term referring to taring and feathering, and other political rebellious acts -- and these were the people who were most outraged over the revolution. Was it mostly due to not being represented? Partly. In the end, there were many reasons to go to war, and this was one of them.

What you heard may be in reference to the Carlisle Peace Commission. Headed by the Earl of Carlisle, it was a contingent of British diplomats who negotiated directly with the Second Continental Congress in the summer of 1778. They basically rolled over on everything and negotiated to meet all the demands the Colonies had at the war's outset, the only difference being that they'd be granted self-rule as British subjects instead of outright independence as a separate country.

Congress ultimately rejected the terms and there's evidence that there was some backlash from the American public who were ready for the war to be over. (A number of local signers of the Articles of Association of 1774-75 in support of Congress had land confiscated or showed up on Loyalist muster rolls in 1779-80.)

But this all happened in the middle of the war, not before it, and the terms were offered at a time when the Americans were in a position that they could actually win the war.

So, yes, Parliament did offer self-rule to the Colonies at one point, but not at the point the Americans had made the demand. Had the offer been extended at the outset, the war may have been avoided.