while there is no way to explicitly set the width, if you really must use tkinter then it is possible to add hacky workarounds to pad these things out. and example of this would be:
import tkinter as tk
from tkinter import font as tkFont
def resizer(event=None):
print("Resize")
widget = event.widget
menu = widget['menu']
req_width = widget.winfo_width()-10
menu_width = menu.winfo_reqwidth()
cur_label = menu.entrycget(0, "label")
cur_label = cur_label.rstrip() # strip off existing whitespace
font = tkFont.Font() # set font size/family here
resized = False
while not resized:
difsize = req_width - menu_width # check how much we need to add in pixels
tempsize = 0
tempstr = ""
while tempsize < difsize:
tempstr += " " # add spaces into a string one by one
tempsize = font.measure(tempstr) #measure until big enough
menu.entryconfigure(0, label=cur_label + tempstr) # reconfigure label
widget.update_idletasks() # we have to update to get the new size
menu_width = menu.winfo_reqwidth() # check if big enough
cur_label = menu.entrycget(0, "label") # get the current label for if loop needs to repeat
if menu_width >= req_width: # exit loop if big enough
resized = True
root = tk.Tk()
var = tk.StringVar()
var.set('First')
option = tk.OptionMenu(root, var, 'First', 'Second', 'Third')
option.bind("<Configure>", resizer) # every time the button is resized then resize the menu
option.configure(indicatoron = False)
option.pack(expand = True, fill = tk.X)
root.mainloop()
this essentially just pads out the first menu item until the menu is big enough. however there does seem to be some discrepancy in the widths reported back by tkinter hence my req_width = widget.winfo_width()-10
offset near the top.
however this will not always be a perfect match size wise, while testing my a space seems to take 3 pixels of width, so it could be 1 or 2 pixels out at any time.