An easy way would be to get OSM maps from a WMS server. Unfortunately public WMS services for OSM maps do not exist. You can have a try with links from here http://wiki.openstreetmap.org/wiki/WMS. With MapInfo all that you need to do is to add a new WMS layer. I suppose you do not want to set up your own WMS server.
There is also a perl script gpx2png.pl. It is actually made for drawing GPX trails on OSm maps - the script downloads the required tiles and stitches them together to form a bigger map. (http://wiki.openstreetmap.org/wiki/Gpx2png). It uses the coordinates of the GPX files to determine the area rendered.
You could make a “fake” GPX file with the coordinates of the area you want a map of and feed that to the script or something,
Some of these options appear to be mentioned on the Oziexplorer page: http://wiki.openstreetmap.org/wiki/Oziexplorer But this wiki page is in urgent need of a rewrite by someone who speaks english. I’d do it myself, but I dont know anything about the software. Perhaps those sections belong on the MapInfo page: http://wiki.openstreetmap.org/wiki/MapInfo (I’ve just added some of the above info there) This page could do with some more attention from an MapInfo expert too.