Sweep and mop the linoleum. Allow the floor to dry. This ensures that the adhesive bonds well.
Pry the shoe molding or quarter-round from around the linoleum floor, using a flat-head screwdriver. Removing baseboards is not necessary.
Glue down any edges or corners of the linoleum that have loosened over time. Use a notched trowel and the flooring adhesive to do this. Wait for the adhesive to dry according to the directions before continuing.
Lay out the asphalt tile over the linoleum. This enables you to position the tiles so that you can cut the ones on either end of the section the same size, using a tile cutter. For instance, the floor looks more aesthetically pleasing if the two end tiles are both one-third of a tile, rather than having a 1-inch strip of asphalt tile on one end of the section and three-quarters of a tile on the other end.
Mark perpendicular lines on the floor to indicate where to begin the tile placement, using a piece of chalk. It's always a good idea to start in the center of the floor or along a wall. Remove the tiles once you decide on the placement.
Read the directions on the flooring adhesive regarding the drying time and cleanup. Use an adhesive that states it is suitable for asphalt tiles. Moisten a cleaning cloth with whatever solution the directions recommend that you use to wipe up excess adhesive. Generally, this is a soapy solution or mineral spirits. This allows you to wipe up excess adhesive while you work and makes cleanup less difficult.
Spread the flooring adhesive on the back of the asphalt tiles. Do this with the notched trowel. Hold it at a 45-degree angle while spreading a thin, even layer of the adhesive -- overlap the strokes if necessary. Wait for the adhesive to become tacky. This usually takes about 30 minutes. Do not allow the adhesive to dry completely before pressing the asphalt tiles into place.
Press the asphalt tiles on top of the linoleum floor. Go over the tiles with a rolling pin to ensure that there are no air bubbles. Do this once you have three or four in place. This ensures that the adhesive bonds with the linoleum. Another option is to walk on the asphalt tiles to help them bond.
Replace the shoe molding once all the asphalt tile is in place. Line up the nails with the existing holes and push the shoe molding back into the wall.