By Lawrence Hurley

WASHINGTON (Reuters) - Conservative U.S. Supreme Court justices on Tuesday expressed skepticism about reviving a lawsuit filed by the family of a Mexican teenager against a U.S. Border Patrol agent who fatally shot the 15-year-old from across the border in Texas in 2010.

In a closely watched case that could affect U.S. immigration actions under President Donald Trump's administration, the court's liberal justices expressed sympathy toward allowing the case to move forward, indicating the justices could be headed toward a 4-4 split. Such a ruling would leave in place a lower court's decision to throw out the civil rights claims against the agent, Jesus Mesa, filed by the family of Sergio Hernandez.

Justice Anthony Kennedy, a conservative who sometimes sides with the liberal justices in close cases and whose vote could be pivotal in this one, voiced doubt about the family's arguments during the court's hour-long argument.

Kennedy indicated the question of how to compensate victims of cross-border shootings is one that the U.S. and Mexican governments should resolve, noting that the border is "one of the most sensitive areas in foreign affairs."

The Supreme Court potentially could delay action on the case to see if Trump's nominee to fill a vacancy on the court, conservative appeals court judge Neil Gorsuch, is confirmed by the U.S. Senate. Gorsuch could then potentially cast the deciding vote. A ruling is due by the end of June.

The justices heard the case at a time that the security of the lengthy U.S.-Mexico border is a hot topic, with President Donald Trump moving forward with plans for a border wall he said is needed to combat illegal immigration.

The case is one of three the justices currently are considering that concern the extent to which the U.S. Constitution provides rights to non-U.S. citizens.

That issue has become more pressing in light of Trump's January order, put on hold by the courts, to block entry into the United States by people from seven Muslim-majority countries and refugees. Trump is preparing a rewritten version of the ban.

The case raises several legal questions, including whether or not the U.S. Constitution's ban on unjustified deadly force applied to Hernandez because he was a Mexican citizen on Mexican soil when the shooting occurred in June 2010.

The court could resolve the case by simply deciding not to apply a 1971 Supreme Court ruling in a case involving federal drug enforcement agents that allowed such lawsuits in limited circumstances. The court has been reluctant in subsequent cases to extend that ruling to other types of conduct.

Kennedy seemed unwilling to take that step, saying the Hernandez shooting would be an "extraordinary case" in which to allow a lawsuit against a federal official.

Liberal justices appeared more willing to examine whether some U.S. rights extend to border areas where the U.S. government exercises a certain amount of authority even beyond the border line, as it does in the culvert where Hernandez was killed.

Justice Elena Kagan said it could be described as a no-man's land that is "neither one thing or another thing."

The incident took place at a border crossing between El Paso, Texas and Ciudad Juarez, Mexico.

The U.S. Border Patrol said at the time that Hernandez was pelting U.S. agents with rocks from the Mexican side of the Rio Grande before the shooting. U.S. authorities have asserted that Mesa shot Hernandez in self-defense.

Lawyers for Hernandez's family disputed that account, saying he was playing a game with other teenagers in which they would run across a culvert from the Mexican side and touch the U.S. border fence before dashing back.

The FBI also said Hernandez was a known immigrant smuggler who had been pressed into service by smuggling gangs that took advantage of his youth.

The family appealed a 2015 ruling by the New Orleans-based 5th U.S. Circuit Court of Appeals throwing out the lawsuit.

(Reporting by Will Dunham)