I am not aware of any proposals to give the American colonies representation in Parliament. Some of the philosophical justification for this lack of representation was the sense of community which English elites felt that they shared with their fellow Britons in the Thirteen Colonies; together they formed an English-speaking trans-Atlantic society united by an Anglo-Saxon heritage, Protestantism, and a belief in commerce and individual liberties. Colonial assemblies were granted jurisdiction over many local matters, while Westminster handled matters of imperial concern, which generally worked well during the first half of the 18th century. Historian Eligha H. Gould argued that shared cultural norms “helped consolidate an emerging sense of belonging to a single British nation among the peoples of the English-speaking Atlantic.” All of this changed with the Seven Years War (1756-1763), after which Britain tried to take a more active control in the running of affairs in the Thirteen Colonies as I'm sure you know. Parliament discovered that the camaraderie that they thought existed with their compatriots in America may did not prevent those same colonists from taking up arms in order to defend against perceived infringements of their liberty.

Even still, there were in fact many in Parliament who supported the plight of the Thirteen Colonies. Many of those who sympathized with the colonists belonged to a parliamentary faction known as the Rockinghamite Whigs, so-called because they were an informal coalition led by the Marquess of Rockingham. The Rockingham Whigs were in opposition for most of the war, which was primarily overseen by the government of prime minister Lord North. North governed from 1770 until he resigned in 1782 following the defeat at Yorktown the previous year.

Perhaps the most famous of the Rockingham Whigs was the influential parliamentarian and philosopher Edmund Burke, who at various points spoke passionately in favour of the right of the American colonists to govern themselves. In a famous speech in 1775, Burke argued that the shared culture and love of individual liberties in both England and the Thirteen Colonies meant that Britons ought to support the push for the colonies to govern themselves:

For, in order to prove that the Americans have no right to their liberties, we are every day endeavouring to subvert the maxims which preserve the whole spirit of our own. To prove that the Americans ought not to be free, we are obliged to depreciate the value of freedom itself; and we never seem to gain a paltry advantage over them in debate, without attacking some of those principles, or deriding some of those feelings, for which our ancestors have shed their blood.

Similar sentiments were echoed by the Duke of Richmond, who spoke in the House of Lords in 1775 and stated that the desire of the Thirteen Colonies to govern themselves was "perfectly justifiable in every possible political and moral sense."

When Lord North resigned in 1782 he was replaced by a pro-peace government run by the Marquis of Rockingham and Lord Shelburne, who concluded the Treaty of Paris with the United States in 1783 which ended the war.

Sources:

Callum G. Brown and W. Hamish Fraser, Britain Since 1707 (2010)

Eligha H. Gould, The Persistence of Empire (2000)

Edmund Burke, "Speech on Conciliation with the Colonies" (1775)

Although he never supported independence and maintained that parliament had the power to pass legislation over the colonies and supported external taxes on trade, William Pitt the Elder did not support parliament's right to pass internal taxes on the colonies due to their lack of representation.

In January 1766 during the Stamp Act crisis, he argued,

"The gentleman asks, when were the colonies emancipated? But I desire to know, when were they made slaves...It is, that the Stamp Act be repealed absolutely, totally, and immediately; that the reason for the repeal should be assigned, because it was founded on an erroneous principle. At the same time, let the sovereign authority of this country over the colonies be asserted in as strong terms as can be devised, and be made to extend every point of legislation whatsoever: that we may bind their trade, confine their manufactures, and exercise every power whatsoever - except that of taking money out of their pockets without their consent."<

His brother in law, Charles Townshend argued that parliament taxes unrepresented groups - East India Company, merchants, stockholders - but Pitt said they are virtually represented as inhabitants with connections to and influence over their representatives as they are landowners and freemen of boroughs. The Americans may be reacting irrationally, but they have been driven to this madness by the injustice perpetrated by Townshend and his supporters.

During the war in 1778 the Carlisle Peace Commission attempted to negotiate with Congress and offered a series of proposals conceeding that the state legislatures had exclusive power to tax their respective colonies and also have the authority to appoint an agent to represent them in parliament, but by that point the treaty of alliance with France had been signed and Congress wanted full recognition of American independence.