Students on campus at St. Gregory’s University, the only Catholic university in Oklahoma. Source: St. Gregory’s University

As Pope Francis advances his call to action against climate change and dependence on fossil fuels, some in the flock are faced with a dilemma. Many U.S. Catholic churches and institutions lease land out to oil and gas companies—and make good money doing it. County documents reveal that dioceses in Texas and Oklahoma have signed 235 leases in oil and gas since 2010, according to Reuters. The pope made a formal call to action in June, saying, "There is an urgent need to develop policies so that, in the next few years, the emission of carbon dioxide and other highly polluting gases can be drastically reduced, for example, substituting for fossil fuels and developing sources of renewable energy." Read MoreCraft brewers get divine inspiration from pope

But at St. Gregory's University, the only Catholic university in Oklahoma, the quandary goes beyond possibly defying the pope's position on climate change. It's also a question of whether to help pay tuition for students who otherwise may not be able to afford it.

First lady Michelle Obama , Pope Francis and U.S. President Barack Obama wave from the balcony during an arrival ceremony at the White House on September 23, 2015 in Washington, DC. Getty Images

The university owns about 800 land leases spread across Oklahoma and Texas, all of which were donated by benefactors over the past two decades. The combined royalties from those leases have averaged half a million dollars each year, or about 3 percent of the university's annual revenue. While that's not a significant percentage of the school's total budget, every dollar counts at St. Gregory's, where 95 percent of its 650 students depend on financial aid in order to pay the school's average, all-inclusive tuition of $30,000.

The royalties provide tuition scholarships for 13 to 15 students every year, according to Greg Main, St. Gregory's president.

"Most students here don't have the income to afford full tuition ... if we're going to serve those students, we need to find a way to supplement (them) by giving them more aid. Those scholarships are where those oil and gas revenues go," Main said. The average family income of St. Gregory's students is $32,106.

To divest or not to divest

Pope Francis' first global encyclical — a papal letter sent to all Catholic bishops — rallies the world against climate change, characterizing the movement as a moral obligation. But nowhere in its 200 pages does it include the word "divest."

Church leaders have continued to lease land for drilling even after the pope's enclyclical, Reuters found. For instance, the Oklahoma City archdiocese allowed three new oil and gas land leases after the distribution of the encyclical.

"This puts us in an awkward position," said Main. "We are duty-bound and will follow the pope's directives. But it's a matter of sorting out what those directives are, because it's not clear right now." Read MoreMass appeal? S&P launches 'Catholic Values' index

The Vatican does not hold direct power over how U.S. Catholic institutions invest their money — that right is reserved for the U.S. Conference of Catholic Bishops (USCCB) and its Socially Responsible Investment Guidelines. The Vatican and the USCCB did not respond to CNBC's inquiries for comment. Any call to "divest" is also missing from the bishops' investment policy. Its only reference to the environment says it "will actively promote and support shareholder resolutions which encourage corporations to act 'to preserve the planet's ecological heritage'" but does not provide specific investing directions.