The　steady-state　laminar　nonisothermal,　incompressible　viscoelastic　fluid　flows　with　the　Jeffrey　model　between　two　counterrotating　(same　direction)　rolls　are　studied　analytically.　The　Jeffrey　model　is　reduced　to　the　Newtonian　model　after　some　appropriate　modifications.　The　new　dimensionless　governing　equations　are　acquired　through　suitable　nondimensional　values.　The　lubrication　approximation　theory　is　employed　for　the　simplification　of　emerging　equations.　The　exact　expressions　for　pressure　gradient,　velocity,　temperature,　and　flow　rate　are　accomplished　whereas　some　quantities　of　interest　such　as　pressure,　coating　thickness,　power　transmitted　by　rolls　to　the　fluid,　and　roll　separation　force　are　computed　numerically　by　using　a　numerical　method.　The　influence　of　emerging　parameters　such　as　the　ratio　of　relaxation　time　to　the　retardation　time　(Formula　presented.),　magnetohydrodynamic (MHD)　parameter　(Formula　presented.)　and　Brinkman　number　(Formula　presented.)　on　velocity　profile,　temperature　profile,　pressure　gradient,　pressure　distribution,　coating　thickness,　power　input,　and　force　of　roll　separation　are　presented　numerically　in　a　tabular　way　and　specific　outcomes　are　demonstrated　graphically.　The　MHD　acts　as　a　regulatory　parameter　for　various　quantities　of　important　interest,　such　as　velocity,　coating　thickness,　and　temperature.　It　is　worth　noting　that　the　coating　thickness　decreases　as　modified　capillaries　number　is　increased.　©　2022　Wiley　Periodicals　LLC.