WASHINGTON (Reuters) - After making major concessions on long-held "fiscal cliff" positions, President Barack Obama and House of Representatives Speaker John Boehner will test the reaction Tuesday of their respective parties in the U.S. Congress and continue talks aimed at further narrowing their differences.

The effort is designed to avert the steep tax hikes and across the board spending cuts set to take effect unless a deal is enacted into law before December 31. Enactment would require a buy-in by the full U.S. Senate and House on whatever Obama and Boehner present to them. Neither Obama or Boehner can be certain yet on how much resistance they might meet.

Though much work remains, the progress contrasted dramatically with previous movement so slow that as recently as Sunday, some Washington insiders saw a 50-50 chance of going over the cliff - which the Congressional Budget Office says would bring on a new recession.

In rapid developments Monday, the two sides came significantly closer to bridging gaps on critical issues such as tax hikes for the wealthy and cuts in Social Security cost-of-living benefits. Those issues have the potential to cause problems politically for both leaders, as Republicans and Democrats start to study them.

Obama and Boehner made the most headway on extending the reduced tax rates originally enacted in the administration of President George W. Bush. Both have agreed to keep the low rates for everyone but the wealthy, but they still differ on who qualifies as wealthy for tax purposes.

Obama, whose definition has for months been taxpayers above the $250,000 threshold, traveled to $400,000 in his latest offer. Boehner was at $1 million, but could move down to $500,000.

Obama also offered a "fast track" process for major tax and spending reforms in the year ahead. A Republican aide who asked not to be identified said that "conceptually," there was agreement to make permanent changes in the tax code, with some of those changes taking effect at the start of 2013 and others at the beginning of 2014.

A comprehensive cliff-avoiding agreement would immediately substitute new and more targeted spending cuts for the indiscriminate slashing of defense and non-defense programs known as "sequestration."

Possible plans to produce cuts in spending for Medicare and Medicaid, the government health insurance programs for seniors and low income Americans respectively, remain to be discussed.

Boehner and Obama have made headway on the politically explosive question of the president's ability to avoid constant battles over raising the debt ceiling, which controls the level of borrowing by the government. Boehner is ready to give Obama a year of relative immunity from conservative strife over the debt ceiling, while Obama is pushing for two years.

Boehner is set to meet with Republicans in the House Tuesday morning and then speak to reporters. Also likely Tuesday is a White House briefing which could shed more light on the work ahead.

The reaction Tuesday from their party allies in Congress may help determine how much further each can go to finish off a deal and how much long it would require to do so.