If you add a column to your date table using this code.
Date (today) = IF ( Dates[Date] = TODAY(), "Today", FORMAT ( Dates[Date], "m/d/yyyy" ) )
Then you can put the [Date (today)] field in your slicer and pick "Today". Whenever the model refreshes (I assume that is daily) this column will get calculated and the row that is marked as "Today" will shift. Since your slicer is set to pick the row that is marked as "Today" your selection moves with it.
Tomorrow the row for 7/13/2019 will be marked as "Today"
If you con't care about seeing the other dates and only care about "Today" moving you can change the code to this.
Date (today) = IF ( Dates[Date] = TODAY(), "Today" )