President Monroe had ties to every major candidate in 1824. John Quincy Adams was his SOS, Henry Clay was Speaker during his presidency, William Crawford was his Treasury Sec while Andrew Jackson was appointed by Monroe as Territorial Governor of Florida. Did Monroe privately have a favorite in this race?
The short answer is no one. Monroe went out of his way to remain neutral in the election, going so far as to delay any appointments when the election was placed in the house so as not to have any accusations of patronage. Generally speaking Calhoun and Adams defended the administration in the election and the other candidates attacked it. Monroe in particular had a very poor relationship with Crawford as Crawford felt he was owed the President's support after Crawford had bowed out of the 1816 election. Crawford went so far as to actively undermine the administration which lead to some very tense movements with Monroe and with his illness Monroe came very close to replacing him. Crawford interpreted Monroe's neutrality as endorsing Adams