Pellegrino only took charge in the summer when replacing Claude Puel, but the change has not had the desired effect with Saints sliding down the table.
Further pressure was piled on Pellegrino following another poor home display against Stoke at the weekend, and it has been suggested he will be sacked at the end of the season even if they beat the drop.
The club’s board do not want to rock the boat now and have faith that Pellegrino will guide them to safety, but a change could be made in the summer with replacements already being lined up.
Jokanovic has worked wonders when guiding Fulham into the promotion mix in the Championship and is said to be their top target, while Ostersund boss Graham Potter has also been mentioned.
Spain National coach Julen Lopetegui has reassured Chelsea striker Alvaro Morata that he still has a chance to earn a call up to the 2018 World Cup squad.