A touchscreen Google Home seems to be hinted at by a new Google job post. Google is now searching for someone to fill the role of Touchscreen Hardware Engineer for the Google Home team and the company has recently listed the job opening over on LinkedIn, detailing what the job will be and what the responsibilities of the person in that role will be. That said, none of the details that Google has pasted in with the listing actually confirm what the product is that is going to be worked on by this person and the team that they will be working with.
Since there aren't any details that confirm the product specifically, everything is left up to a little bit of speculation at this point, but Google has been rumored to be working on a Google Home unit with a touchscreen for a little while now so it could launch a product that would be able to compete with Amazon's Echo Show. While the existence of this job alone doesn't confirm that a touchscreen Google Home is in development, some information about the role mentions that the touchscreen hardware engineer would be working on "next-generation" Google hardware and that the person would need to help enable the best multi-touch user experience.
Further reading points to things like selecting controllers and manufacturing vendors for the sensors as well as defining sensor patterns. Taking these responsibilities into consideration it's a safe bet that this job is in direct reference to the development work on a Google Home product that will have a touchscreen attached to it. The job posting also says that the role will include leading the touch module development from concept to mass production, which means that development on such a product hasn't started yet and it's likely to be a while before Google launches a competitor to the Echo Show. That is unless Google has plans to launch more than one product of this kind and there is another that's already in development. With the job posting now out there it looks like Google really is on its way to developing a new Google Home product that comes with a display, but there's no indication of when it might be ready.