The use of TP140V high-grade steel casing in deep and ultra-deep wells imposes more stringent requirements on the anchoring stability performance of packer slips. To address this, a mathematical model for evaluating the anchoring stability of slips was derived based on mechanical equilibrium and the casing's resistance to external collapse strength, clarifying the depth range within which the slip embedding into the casing remains stable. Finite element analysis was employed to investigate the mechanical behavior of the casing and slips under different slip structural parameters. The research results indicate that increasing the slip body inclination angle, tooth edge distance, and tooth spacing, while reducing the cylinder angle, can effectively improve the uniformity of slip teeth embedding into the casing. The stress on the slips is positively correlated with the slip inclination angle, tooth diameter, and tooth edge distance, and negatively correlated with the tooth body inclination angle and cylinder angle. These relationships can guide the selection of appropriate values to enhance the load-bearing capacity of the slip body during slip design. The bottom of the slip body is a vulnerable point for slip teeth and should be a key monitoring location for slip failure. The findings provide theoretical basis and guidance for the structural design of slip teeth embedded in high-grade steel casings.