Round purses with drawstrings have been around for a very long time indeed, those produced in the 17th century with flat bottoms tend to be known as gaming purses. Although playing cards had been around for a couple of centuries these purses became fashionable at much the same time as the vogue for card games, and were used to carry money or gaming counters. The still life painting of the Five Senses by Baugin (1612-1663) shows a very plain version of this type of purse on a table
I have embedded links to museum examples of many of the surviving purses. These purses are heavily embroidered, and are often described as French on the basis that sometimes the decoration is of fleur de lys. The original ownership of some purses can be seen in the bases of the purses, which are often decorated with the coat of arms of their owner. The museum at Versailles for example has two purses, one with the coat of arms of the Marquis de Louvois (1641-1691) and the other with the arms of Phillippe, Duc de Orleans (1640-1701) brother of Louis XIV and brother in law to Charles II of England. These two links show the flat bottoms of the purses with their coats of arms. There is a very similar purse in the Museum of London which dates from the beginning of the 17th century.
The Victoria and Albert Museum owns several examples, including a much simpler one in green velvet, with a copper gilt cord decoration. The Metropolitan Museum in New York also has several examples, including one in needlepoint. In most cases the purse has a fabric outer, usually covered, in the surviving examples, with goldwork embroidery, they are often lined with silk, the stiff circle of the base being of leather. The purses that survive may well do so precisely because of the decoration. The way the drawstring braid is inserted can be seen clearly in one of the photos of this example from the V&A collection which has the arms of the Cardinal Duke de Matignon . The Met describe the finials on the drawstrings as wooden beads covered with metallic thread.