CAIRO — With the Muslim Brotherhood pulling within reach of an outright majority in Egypt’s new Parliament, the Obama administration has begun to reverse decades of mistrust and hostility as it seeks to forge closer ties with an organization once viewed as irreconcilably opposed to United States interests.

The administration’s overtures — including high-level meetings in recent weeks — constitute a historic shift in a foreign policy held by successive American administrations that steadfastly supported the autocratic government of President Hosni Mubarak in part out of concern for the Brotherhood’s Islamist ideology and historic ties to militants.

The shift is, on one level, an acknowledgment of the new political reality here, and indeed around the region, as Islamist groups come to power. Having won nearly half the seats contested in the first two rounds of the country’s legislative elections, the Brotherhood on Tuesday entered the third and final round with a chance to extend its lead to a clear majority as the vote moved into districts long considered strongholds.

The reversal also reflects the administration’s growing acceptance of the Brotherhood’s repeated assurances that its lawmakers want to build a modern democracy that will respect individual freedoms, free markets and international commitments, including Egypt’s treaty with Israel.