Set one piece of linoleum that matches the existing floor over one of the holes to get an idea of how much linoleum is needed to cover the hole.
Lay the piece of linoleum on two pieces of cardboard set on top of each other or on one piece of plywood to protect your work surface. Use a utility knife to cut a patch of the linoleum piece that is large enough to cover the hole and extend a couple of inches. Don't cut a section out of the middle of the linoleum piece.
Place the patch on the hole and reposition it until its pattern aligns with the flooring's pattern. Tape the patch in place with masking tape.
Place a straightedge on top of the patch. Position the utility knife against the straightedge so you can make vertical cuts. Cut down and through the patch and the linoleum section that includes the hole. Repeat the cuts around the entire patch and through the linoleum flooring around the hole.
Remove the patch and the linoleum underneath. Use a flat-head screwdriver or putty knife to lift up the glued linoleum if necessary. Discard the section with the hole.
Put the patch into the opening to ensure it matches and fits well. Remove the patch.
Apply linoleum adhesive directly to the floor below the linoleum by using the product's squeeze applicator or a V-shaped trowel.
Press the new section of linoleum into place on the adhesive. If the linoleum adhesive oozes out the sides, clean it up with a damp cloth.
Squeeze out pockets of air in the flooring area with a hand roller or rolling pin.
Lay a heavy telephone book or other heavy, flat object over the linoleum patch until the adhesive sets.
Repeat the procedure for each hole in the linoleum.