"Negotiations behind closed doors are likely continuing and some type of deal to avoid a major fiscal crisis is still possible,'' Esiner said. "However,every passing day increases the risk of going over the cliff, which remains positive for the dollar in the near-term.''
The euro was last nearly flat at $1.3185. Offers were cited above $1.3240. It hit an eight-month high of $1.33085 last Wednesday after speculators cut bets against the currency in recent weeks.
The dollar index was up 0.1 percent at 79.66. Against the yen, the dollar was last up 0.7 percent on the day at 84.83 yen. Chartists said the dollar needed to overcome 85.05 yen, its 200-week moving average, for it to sustain further gains.
"There has been some pretty significant yen selling all through the night and into this morning,'' said Peter Kinsella, currency strategist at Commerzbank.
"It is very noticeable we have not seen any retracement or dip in dollar/yen at all. The market is really saying they are convinced on yen weakness and that is what we are going to see for the remainder of this year and in the course of next year.''
The U.S. currency hit a 20-month high of 84.62 yen last Wednesday as the yen fell after a landslide election victory for Abe's Liberal Democratic Party.
Abe, set to become Japan's prime minister on Wednesday, has called for aggressive monetary stimulus by the BOJ to beat deflation and this has dragged on the yen.
However some strategists said further dollar gains against the yen could be limited, given the extremely bearish positioning on the Japanese currency.
Careening Toward 'Cliff'
The yen also fell against the euro. The single currency was up 0.7 percent on the day at 111.85 yen, not far from a 16-month high of 112.59 yen, hit on Dec. 19.
Strategists said developments on the Italian elections and Greece could see it grind higher in thin year-end trading.
However, if an impasse over the fiscal cliff deepened, investors could sell it for the more liquid dollar.
Some U.S. lawmakers voiced concerns on Sunday that the countrywould go over cliff, triggering tax increases and spending cuts early next yearthat could push the economy back into recession.
Focus has shifted to Congress acting after Jan. 1.
Marc Chandler, global head of FX strategy at Brown Brothers Harriman in New York, believes the U.S. government will go over the cliff in January,but a deal will be eventually worked out.