Half of the people questioned in a CNN/Opinion Research Corporation survey say they'd support such a decision, with 49 percent opposed.
But support for a troop buildup of that size is greater than the 45 percent of the public who support the war in Afghanistan. The survey indicates that 52 percent oppose the war.
"The war is unpopular and previous polls have shown that Americans oppose sending more troops in the abstract," says CNN Polling Director Keating Holland. "But it may be a different story when Americans are confronted with an actual decision, by the Commander-in-chief, on a military matter. Previous presidents have seen a 'rally effect' - at least temporarily - when they have made command decisions like this one."
The poll's Tuesday morning release comes just hours after Obama met Monday night with his national security team on Afghanistan and neighboring Pakistan. This was the ninth meeting of the president's war council to consider whether to send more troops to Afghanistan, as requested by the U.S. commander on the ground there.
White House Press Secretary Robert Gibbs said Monday that Obama won't announce his decision on a troop deployment this week, saying that the "first possible time would be sometime next week."
Sources told CNN that one option presented to Obama calls for sending about 34,000 more U.S. troops to Afghanistan in addition to the 68,000 already committed to the country, with other options involving variations of that plan.
According to the poll, if the president decides to send a smaller number of troops than 34,000, 56 percent would oppose the move, with just over four in 10 backing the decision.
"Support for a smaller troop buildup drops most among Republicans," Holland notes. "They appear most wary of doing things by half-measures in Afghanistan."
The survey also indicates that two-thirds of Americans say things are going badly for the U.S. in Afghanistan. That's up 11 points from March, when 55 percent said things were going bad.
People questioned in the poll were also asked about the war in Iraq. Opposition to that conflict remains high, with 62 percent saying they oppose the war. But 57 percent say sending more troops into Iraq in 2007, known as the "surge," was a success.
The CNN/Opinion Research Corporation poll was conducted November 13-15, with 1,014 adult Americans questioned by telephone. The survey's sampling error is plus or minus 3 percentage points.
Follow Paul Steinhauser on Twitter: @psteinhausercnn